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Proceedings of the International Conference on Mathematical Analysis and its Applications, Kuwait, 1985
Edited by
S. M. MAZHAR, A. HAMOUI and N. S. FAOUR
Kuwait University, P.O. Box 5969 Safat, Kuwait
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Copyright (C) 1988 Pergamon Press pic. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. First edition 1988 Library of Congress Cataloging in Publication Data International Conference on Mathematical Analysis and its Applications (1985 : Kuwait, Kuwait) Mathematical analysis and its applications. (KFAS proceedings series ; v. 3) 1. Mathematical analysis—Congresses. I. Mazhar, S. M. II. Hamoui, A. (Adnan) III. Faour, N. S. (Nazih S.) IV. Title. V. Series. QA299.6.1565 1985 515 88-4055 British Library Cataloguing in Publication Data International Conference on Mathematical Analysis and its Applications (1985 : Kuwait) Mathematical analysis and its applications. 1. Calculus I. Title II. Mazhar, S. M. III. Hamoui, A. IV. Faour, N. S. V. Series 515 ISBN 0-08-031636-0
In order to make this volume available as economically and as rapidly as possible the author's typescript has been reproduced in its original form. This method unfortunately has its typographical limitations but it is hoped that they in no way distract the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
ORGANIZING COMMITTEE
Professor Fuad S Mulla Professor S M Mazhar Dr Adnan Hamoui* Dr Waleed Deeb Dr Nazih S Faour
Dr R Younis Dr M N Al-Tarazi Dr M Al-Zanaidi
* Also representative of the Kuwait Foundation for the Advancement of Sciences
EDITORIAL COMMITTEE
Professor S M Mazhar
Dr Adnan Hamoui, Dr Nazih S Faour
ORGANIZING COMMITTEE
Professor Fuad S Mulla Professor S M Mazhar Dr Adnan Hamoui* Dr Waleed Deeb Dr Nazih S Faour
Dr R Younis Dr M N Al-Tarazi Dr M Al-Zanaidi
* Also representative of the Kuwait Foundation for the Advancement of Sciences
EDITORIAL COMMITTEE
Professor S M Mazhar
Dr Adnan Hamoui, Dr Nazih S Faour
PREFACE
An international conference on "Mathematical Analysis and its Applications", sponsored jointly by Kuwait University and the Kuwait Foundation for the Advancement of Sciences, was held in Kuwait from February 18 to February 21, 1985. It was attended by a large number of mathematicians from all over the world. Twenty one invited talks were delivered by eminent mathematicians and about 53 research papers were presented in the form of short communications.
The present volume contains the texts of some of the invited talks and research papers presented at the Conference. All articles appearing in these proceedings were duly refereed with the exception of invited talks.
It is a pleasure to express, on behalf of the Department of Mathematics, sincere thanks to Kuwait university and the Kuwait Foundation for the Advancement of Sciences for providing necessary funds towards holding the conference and the publication of its Proceedings.
We acknowledge our indebtedness to Dr Bader Al-Saqabi, ex-Chairman of the Department of Mathematics, for placing resources of the Department at our disposal. We also express our gratitude to the invited speakers for giving stimulating talks at the conference, and to all participants for making the conference a success.
Finally the staff of Pergamon Press merits our warm thanks for publishing the proceedings.
S M Mazhar Adnan Hamoui Nazih S Faour
1 May 1987
vii
AN ESTIMATE FOR THE RATE OF CONVERGENCE OF A GENERAL CLASS OF ORTHOGONAL POLYNOMIAL EXPANSIONS OF FUNCTIONS OF BOUNDED VARIATIONt
R. BOJANIC* Depar tment of Mathematics , Ohio State University,
Columbus, OH 43210, USA
1. Let p (x) be a sequence of o r thogona l polynomials genera ted by a
non-nega t ive weight f u n c t i o n ω(χ) on [ - 1 , 1 ] and 1
hn = [ ω ( * ) ρ * ( £ ) dt. (1 .1 )
- 1
We denote by
%(f) = (]/hn) \ "it) f(t) pn(t) at , -1
tt= 0,1,2,... , the Fourier coefficients of / with respect to p (x) and by
lak(f)pk(x) (k> 0) ,
the Fourier expansion of /. The n-th partial sum of this series will be denoted by
η Sn(f,x) = J ak(f)pk(x).
k=o In 1952, G. Freud proved in [11] that under certain hypotheses about
the weight function ω(χ) it is possible to conclude that lim Sn(f9x) = fx) (1.2)
ft-*oo
Invited talk. The author is grateful to the Kuwait University and the Kuwait Foundation for the Advancement of Sciences for the support of his participation at the Conference on Mathematical Analysis and its Applications.
1
2 R. Bojanic if / is a function of bounded variation on [-1,1], continuous at the point #€(-1,1). Freud1 s result in its simplest form can be stated as follows.
Suppose that the weight function ω(α?) satisfies the following conditions on (-1,1):
(i) ω(χ) < M\-x2)~A ,
( i i ) \p x)\ <M\-x2)~BhU2 .
Suppose further that / is a function of bounded variation on [-1,1]. Then (1.2) holds at every point x€ (-1,1) where / is continuous.
This result is clearly the analog of the well-known Dirichlet-Jordan test for the convergence of ordinary Fourier series of 2π-periodic functions of bounded variation.
We shall consider here the problem of estimating the rate of convergence of the sequence of partial sums s (f *x) when / is a function of bounded variation on [-1,1], not necessarily continuous at the point x € (-1,1).
In order to obtain the rate of convergence we have to assume that, in addition to (i) and (ii) , the sequence of polynomials p (x) satisfies the condition
i f i chU1
(iii) J ut)Vnt)dt\ <-£- . -1
Our main result can be stated as follows. THEOREM 1. Let p (#) be a sequence of orthogonal polynomials on [-1,1] generated by a non-negative weight function (ύ(χ) and let h be defined by (1.1). We assume that for every χ € (-1,1) and n — 1,2,... conditions (i), (ii) and (iii) are satisfied.
If / is a function of bounded variation on [-1,1] then n
K(f,*)-t(/U + 0)+/(,-0))|<££> 1 V^Y^'ligJ + \ | / ( * + 0) -f(x-O) | \Sn$x,x)\. (1.3)
Here, g is defined by 'fit)-f(x-o) if - i < t < x < i
gxW = 0 if t = x (1.4) ft)-f(x + 0) if -\<x<t<\
and ψ is the special function ». X -1 if t<x
tyxt) = s i g n ( t - x ) = \ -0 if t=x ( K 5 )
1 if t > x .
Orthogonal Polynomial Expansions of Bounded Variation 3 Also , C(x) > 0 for x £ ( - 1 , 1 ) and V (f) i s the t o t a l v a r i a t i o n of the func
t i o n / on [ a , b ] .
I f f i s , i n a d d i t i o n , con t inuous a t the p o i n t x € ( - 1 , 1 ) , i n e q u a l i t y
(1 .3) becomes
\Sn(f,x)-f(x) <— I V M + w . (f) · n n u x — ( i +x) j \ k=\
Theorem 1 shows that the convergence of the orthogonal expansion of a function of bounded variation, at a point of discontinuity, depends essentially on the behaviour of the orthogonal expansion of the special function ψ (t) = sign(t-x) at the point t=x .
It is easy to see that condition (iii) is satisfied if and only if the Fourier coefficients of ψ with respect to pn(x) satisfy the inequality
n h n
It is interesting to observe here that this condition, or condition (iii), which we had to add to Freud's conditions (i) and (ii) in order to obtain an estimate of the rate of convergence, is itself related to the
2 rate of best L -approximation of the function ψ by polynomials. Using orthogonality property of the sequence p (x) we have, for any polynomial Q of degree < n - 1 ,
n 1 χ W * « ' s ί ω(*) **(*>?«(*><** -1 f ω(*)(φΛ(ί)-βκ_ι(*))ρ2(ί)4ί .
Hence -1
\l/2 / 1 vl/2
Καη^χ) < (J ω(*) (**(*)-Vi(t)) dt) (l ^Pnt)dt) -1 -1
Taking infimum over all polynomials of degree <n- \ we find that
Κ\αη(*χ)\ < Α ^ 2 £ ^ 2 ) ( ω . « Γ ) ,
(2) where En (ω»Ψχ) is the constant of best L2-approximation of ψ by polynomials of degree < n - l . Thus, condition (iii) will be satisfied if
£(2)(ω,ψ ) < - . un ^>ψχ' ^ η
Theorem 1 is a very useful tool for the study of convergence properties of specific orthogonal expansions of functions of bounded variation.
4 R. Bojanic A particular and yet sufficiently general system of orthogonal polyno
mials to which Theorem 1 is applicable is the system of Jacobi polynomials lPn ' (x) generated by the weight function
ω(χ) = (1 -xf (1 +x)3
(a ß) fn + a\ and normalized so that P * ( 1 ) = I I . We have here n \ n /
Α ( α , β ) 2 α + 3 + 1 Γ ( η + α + 1 ) Γ ( η + β + 1 ) η 2η+α+3+1 Γ ( η + 1 ) Γ ( η + α + $ + 1 )
and, as i t i s easy to s e e ,
Ί . , ( a , 3) a + 3 , , ,v l im nh = 2 . (1 .6 ) n-»<*>
In case of Jacobi polynomials, properties (i), (ii) and (iii) can be easily established. An estimate for the rate of convergence of the sequence Sn(4> >%) to zero, following a suggestion by R. Askey, can be obtained as follows. Since x is an interior point of [-1,1], we can first use the equiconvergence theorem for Jacobi series (see [12], Th.9.12, p.244) to conclude that lim Sn^x>x) = 5 Vx (χ + °) + Ψχ(*-°))= ° > n ->°°
which is equivalent to the statement that oo
Sn^x>*)=- l α,ίψ )P,(a'e)
It follows then that oo
α, (ψ ) P^'p' (x)
and an estimate for S (ψ , χ) can be obtained by using asymptotic properties of Jacobi polynomials.
We have the following estimate of the rate of convergence of the Fourier-Jacobi expansion of a function of bounded variation.
THEOREM 2. Let Ϊ? ' (#)f be the sequence of Jacobi polynomials and let / be a function of bounded variation on [-1,1]. I_f_ a > - anc* 3 > ~ 2» we have
I C(a,3;a0 V x + (\-x)Ik , v ^ ) γ (q) ?„(a,3)(f,Ä) -|(f(x + 0) +f(x-0))
+ k=i M(a,$;x) ,
]f(x + 0) -f(x-O)
Orthogonal Polynomial Expansions of Bounded Variation 5 In Section 2 we shall give a brief survey of results related to Theorem
1 which have been obtained so far. In Section 3 we shall deduce Theorem 1 from a general estimate of the rate of approximation of a function of bounded variation by certain integral operators. Finally, in Section 4 we shall give a proof of Theorem 2.
2. An estimate of the rate of convergence of the partial sums of the Fourier series of a 27T-periodic function of bounded variation was given by R. Bojanic in [1]. In that paper it was proved that
n Sn[f,x)-\(f(x + 0)-f(x-0))\<l y V*,k (gx) ,
where g (t) = f(x+t) +f(x-t) -f(x + 0) -f(x-O). An extension of this result to certain functions of generalized bounded variation in the sense of Waterman was published in [7] by R. Bojanic and D. Waterman.
The rate of approximation of a 2iT-periodic function of bounded variation by Nörlund-Voronoi means of its Fourier series was studied in [4] by R. Bojanic and S.M. Mazhar. It was proved in that paper that
n . \Vn(f>x) ~ \ (/(* + 0) +/(x-0))| <^- £ -PkVQ
7[ k(gx) . k= o
n Here p is a non-increasing sequence of positive numbers, P = Y P7 -> °°
n n ^ = 0 κ
(n-»°o) and n
k=o
is the Norlund-Voronoi mean of the sequence of the partial sums of the Fourier series of /. This result was recently extended to triangular matrix means by S.M. Mazhar [13]. A summability result of a different type was obtained by Bojanic and Mazhar in [5]. In that paper the authors have obtained an estimate of the rate of approximation of a 27T-periodic function by integral operators of the form
II
Kn(f,x) =i j /(x.t) kn(t)dt
under the hypothesis that k (x) is a 2iT-periodic, even and integrable function satisfying the following conditions:
π - ί k (t)dt = l
0
\k (t) | <An , 0<£<π
6
and
R. Bojanic
k (u) du [nt
v l + o t ' a > - i
If these conditions are sa t i s f i ed , then
-X (/.*)- l(/<*+0>+/<*-0)) < - ^ J kavl/k (g).
li *- | 1 + Ot '—-' X
The first result of this type for Fourier-Legendre expansion of a function of bounded variation on [-1,1] was obtained by R. Bojanic and M. Vuillemier in [6]. If
n
is the n-th partial sum of the Fourier-Legendre series of / with
1 ak(f) = (fc + j) J f(t)Pk(t)dt,
then
Sjf.x) -±(f(x + 0) + /(*-0). 28(1 -xz) 2%-3/2 n
v> x - (i-x) Ik 1 v (i-+ ί±Ζ^— l/(*+ <>)-/(* 0)1 ,
x - (i+x)/&
where # (t) is defined by (1.4). It should be mentioned also that R. Bojanic and Z. Divis have obtained
in [3] an estimate for the rate of convergence of the partial sums of a special eigen-function expansion of a function of bounded variation. The proof of that result was further simplified by Z. Divis in [10].
Estimates of the rate of approximation of functions of bounded variation by certain polynomial operators were obtained by F. Cheng [8]. For Bernstein polynomials
n t-k Bnu.*)-i /(£)(;;) **<-*>"-
k=o he obtained in 1983 the following result.
If f is a function of bounded variation on [0,1] then n
I Λ ( ~Λ Ax) ΆΚΧ) sp x < ) V
n L x
x+ (ι -x) NU
fc=i zlJk <**>
Hx) \f(x + 0) - f(x-0)\
If, in addition, / is continuous at # 6 ( 0 , 1 ) , then
Orthogonal Polynomial Expansions of Bounded Variation
Bn(f,x)-f(x) Ax) v x + \-x)lA
L k = i
V -xiA (f)
A similar result for Szasz-Mirakyan operator - / \ k
„in \ -nx V Jk\ (nx) k=0
was published by F. Cheng in 1984 (see [9]). We should mention finally that R. Bojanic and F. Cheng have obtained
results of this type for Hermite-Fejer interpolation based on the zeros x, = cos ί ττΐ ,fe= Ι,.,,,η of the Chebyshev polynomial T (x) = cos(narccosi), at points x€ (-1,1) where f(x + 0) = f(x-O) (see [2]). The main result of that paper, however, is a divergence theorem which states that the Hermite--Fejer interpolation polynomials of a function of bounded variation diverge at the jump points x € (-1,1) where f(x +0) Φ f(x-Q).
3. In this section we shall prove a theorem of general nature which gives an estimate for the rate of approximation of a function of bounded variation by certain integral operators.
Let Kn(x, t) be a continuous function with respect to x on [-1,1] and an integrable function with respect to t on [-1,1]. For a function of bounded variation on [-1,1] we define
1 Ln(f9x) = \ Kn(x,t)f(t)dt.
-1 THEOREM 3. Suppose that the function K (x,t) satisfies the conditions
(a) J V*>*) dt
and
(b)
(c) !*„(*.«>
x+(1-x)In
x-(1+x)In
B(x)
\K (x9t) \dt < A(x)
du n(x - t)
K (x 9u) du Hx) n(t -x)
for every
for every
- 1 < t < x < 1
-1 <x <t < \
If / is a function of bounded variation on [-1,1], then
R. Bojanic
y/.x)-i(/(«+o)+/(x-o))|<^5; /„ ; ; ; : ; ; : ^ + ||/(^-ο) + / u + o ) | μη(ψΑ,Α)| . (3 .0
Here, g and ψ are defined by (1.4) and (1.5) respectively. This theorem, in a slightly different form, was stated in [6] and a
proof was published in [3]. We shall give here, for the sake of completeness, a short sketch of the proof.
The proof of Theorem 3 is based on the following inequality for the Riemann-Stieltjes integral
b b
| J f(t)d(g(t))\ < ( \Μ\άνΙ(ρ) . a a
Here, f is a continuous function and g a function of bounded variation on [a,b].
Proof of Theorem 3. We have 1
Lnf9x) = j Knx9t)gx(t)dt
- 1 x 1 + f(x~0) [ K x9t) dt + fx-0) K[x9t)dt
- 1 x where g i s def ined by ( 1 . 4 ) . S i n c e , by ( a ) ,
Jy*. t) dt = 1 ,
-1
we can rewrite the preceding relation as follows: 1
Lnf>x)-\ ( /<*+0) +f(x-0))= j Kn(x9t)gx(t)dt - 1
X 1
Hence
+ \ (/(ar + 0) - f(x-0))(- J Knx9t)dt +^Knx9t)dt - 1 x
1
in(f>x)-\ (f(x+0))+f(x-0)) = J Kn(x9t)gx(t)dt -1
+ | ( / ( χ + 0 ) - / ( χ - 0 ) ) £ η ( φ α ? > χ ) ,
where ψ is defined by (1.5). Thus, in order to prove Theorem 3 we have only to show that for any function g of bounded variation on [-1,1], which is continuous and vanishes at the point x € ( - 1 , 1 ) , we have
Orthogonal Polynomial Expansions of Bounded Variation
j Knx9t)g[t) dt C(x) v x + (\-x)/k
<~n~ l vx-tx+x)/k Mi
provided K (x 9 t) satisfies hypotheses of Theorem 3. We have 1 ^Kn(x9t)g(t)dt =
x-(\+x)/n x+(\-x)/n i
ί * J * J _1 x-(\+x)/n x+(\-x)/n
= Ang,x) +Bn(g,x) + Cn(g9x) .
Since g(x) = 0, we have first, by (b) x+ (1-x)In
\ \Knx,t)\ \g(t)-g(x)\dt Bng,x)
Next, let
x-(1+x)/n x + ( 1 -x)In
<A(x) V (g) . x - ( 1+x)In
Κ(χ>*) = j Knx>u)du
and let y = x -(] +x)In . We have then, by partial integration y
An(g,x) = gt)dAn(x9t)
v = gy) hn(x9y) + j A n ( * , t ) ^ ( t ) .
Since g(x) = 0 , we have y y
Ang,x) = (g(y)-g(x)) \ Kn(x9t)dt+ j An(x91) dg (t) .
Using inequality (3.2) and the first of hypotheses (c) we find that
\An(g9x)\ <VXy(g) · B(X) + i n(x-t) v t y ' i(x-y) _!
Using partial integration again we find that x - (1 +x) In
\Ajg,x)\ < 7 (g) , .. + - ^ - f - ^ — - dt - l n(x - 1) -1
(x-t)2
Replacing t in the last integral by x- (\+x)/t we find that
10 R. Bojanic
x-(1+x)In
dt = — — f Vx , t J . ,,.(g) dt 1 +x J x - ( l + x ) / t v y / ^ >
.1 < * " * > 1
It follows that
n - i fe + i
= 7 ^ Σ J C o ♦«)/*<'><*' fe = i k
<ΊΤΐ Σ v*-o+x)/k g)' k = l
, . /4B(a:)\ \ £ x
k=i Using the same arguments we find that
n /bB(x)\ l
\C "·'" ' ' ' \1 -xzl n
k=\
From inequalities for 4 (^9x),5 (g,x) and C (g>x) follows that n
^4B(a:)\ i γ a; + ( 1-ar
and (3.4) follows since n
x + (\-x)Ik x - (\-x) In v-< x + i-x)/Kt x - \\-x) in ) V (g) > n V (q). L x - (\+x)lk V y / x - (\+x)/k
The proof of Theorem 3 now follows immediately if we observe that the function g defined by (1.4) is of bonded variation on [-1,1], continuous at t = x and q (x) = 0 .
yx Proof of Theorem 1 In order to deduce Theorem 1 from Theorem 3 let us observe first that the n-th partial sum of the series
I ak(f) PkM (k > 0)
can be written in the form
S.
where
„(/,*) = Kn(x,t)ft)dt ,
Orthogonal Polynomial Expansions of Bounded Variation 11
Kn[x9t)=o>(t) I (l/hk)pk(x)pk(t) . k=o
Consequently, in view of Theorem 3 we have only to show that conditions (a), (b) and (c) are satisfied if conditions (i), (ii) and (iii) are satisfied.
We have first, by orthogonality 1 1
J Kn(x,t)dt = (\/hQ) pQ(x) I u(t)pQ(t)dt. - 1 - 1
Since p (x) is a constant, we have ° 1 1
| Kn(x9t)dt = (1A 0 ) j (*t)p2Qt) dt = 1 .
- 1 - 1 So, condition (a) i s sa t i s f i ed .
Condition (b) i s also sa t i s f i ed since
.r+ ( 1 -x) In
K x9t) x-(i+x)In
dt
x+(l-x)In \ ω(ί) x-(l+x)In
\Uhk)pkx)pk[t) k=o
dt
(x+(l-x)ln Y ' 2 / l / n
<ij φ)άή | ^ ) n ( i / y p ^ ) ^ ) | dt \x-(i+x)In I V _ 1 \k = 0
tf*
/ * + ( i - x ) / n \ 1 / 2 / n 2, , V / 2
\x-(i+x)/n j \fc=0 K J
For n > 2 we have, by (i) , ω(£) < Mmax (l - τ2)'Α : - (l -x)/2 < τ < (l + x)l2 = Mx)
for t G 1 + x , 1 - x
x — , 1 + —-— On the other hand, by ( i i ) ,
p2k(x)/hk<M2(l-x2)-2B.
Consequently, we have x + (l-x)In
u*.t)\dt<^y2w-*rB I«**)1* x- ( i+x ) In
and (b) follows.
12 R. Bojanic The proof that conditions (ii) and (iii) imply (c) is based on the
Christoffel-Darboux formula which we will write in the following form
£ (\/hk) vkx) pk( Ύν
* = V ^ i 0*0 P (£) - p _,_. (£)p (a:)
k=Q V n+ 1 n x - t
where |γ | < 1 . Using this formula we find that for -1 < t < x < 1
-1 ^ n +\ n i ω(τ)
p , Xx) p (τ) -ρ , ,(τ) ρ (x) άτ ,
Since \/(χ-τ) for fixed x E (-1,1) is an increasing function of τ on [-1,1], we find by the second mean-value theorem that
t Ύ * 1. V*' T ) d T = 7 f = ^ ( P n + l f a ) ί ω(τ)ρη(τ)<ίτ -1 ίΤΓΤΓϊΓ
\J n+\ n
P„(«) j ω(τ)ρη+1(τ)^-Λ where - K e < t .
Using now conditions (ii) and (iii) we find out t
\ V*>T) άτ l*MC(\-x2)
n(x - t)
and the first of conditions (c) follows. The proof that the second of conditions (c) is also satisfied is similar.
4. Proof of Theorem 2. In order to prove Theorem 2, we have to show that Jacobi polynomials .P (x) satisfy hypotheses (i), (ii) and (iii) of Theorem 1 if a > — » 3 > - a n d w e have to estimate S * (ψχ9χ) . Condition (i) is clearly satisfied.
To see that condition (ii) is satisfied we observe that for x € (0,1) and a > - 2 > 3 > - ^
Ρ(α'β)(χ) n v '
M
(1 -x # J_
2xa/2+ l/h ' /R (4.1)
[see [14], p. 16 7). If x € ( - 1 , 0 ) we have likewise
3 ( α , β ) , (*) | < (i-xzf/2+1/k /n
( a R oi + R Since lim nh * =2 , we see that condition (ii) is satisfied with YL
B = max (a/2 + 1/4, 3/2+ 1/4) and M sufficiently large. To see that condition (iii) is also satisfied, observe that
(4.2)
Orthogonal Polynomial Expansions of Bounded Variation
( i - t ) a ( . + t ) P p n( a ' e ) ( t )
^ i ( c - ) a + i o + * ) ß + i e v , ß + i ) ( * ) ) · Since β + 1 > 0 i t follows that
13
( i - i ) a ( i + t ) M a > ß ) ( t ) d t
= £(.-*)β + 10**)Β + 1Ρ<ο + 1· β + 1>(*) Using inequalities (4.1) and (4.2) we find that
(4.3)
; ι - * ) α + 1 ( ι + * ) 3 + 1 p(a+l,3+l) χ)
n-\
< c ( ' _ + J U T T W I - P (i-xf+1( i + * ) e + 1
< 2α+β(ΐ-^)α/2 + ΐ/ΐ,(ΐ+α;)β/2 + 1 / ν ^ ( α ' β ) ν / 2
V n j
)
holds uniformly in x € [-1,1]. Inequality (iii) now follows from this inequality and (4.3).
We shall next study the rate of convergence of the Fourier-Jacobi expansion of the special function ψ (t) = sign (t - x) at the point t = x.
We have
* ί α · β ) < ν > - Σ "*<**> *£α·β )<*>. fe=o
where du
-1
+ ί (l-u)a(l + M) ßP ( a' ß )(u)du
For fc > 1 we have, by orthogonality of Jacobi polynomials, x
\ (α'β) α Λ ) = - 2 j o - " ) a ( i + " ) ß ^ a , ß ) ( " ) d u , or, using (4.3)
-1
ί '^Α)^!' -^ 1 ! 1 - )^ 1 ^ 1 ·^ 1 ^ We have, therefore
14 R. Bojanic 5„(α'3 )(Ψτ,*) = αη(ψ ) 7ί ^ 0 ν τ X '
+ (ι_χ)^ι(1+χ)3+ι y (λΑ^,β)^ΐρ(α+ι,3+ι)(Λ) ρ(α, 3) (t) ? fe-i
k=\ where
x l ( Ψ χ ) = ( \ ( α , β ) ) _ 1 ( - 1 l - U ) a ( l + u ) ß d M + ( l - u ) a ( l + M ) ß d u ) .
-l x
Since ψ^ is a function of bounded variation, we have, by the equicon-vergence theorem mentioned earlier, for every t € [-1,1] and je G ( — 1,1), Ψχ(*)= signt-x) = aQ(\px)
+ (l-x) a + 1 (l +x) 3 + 1f (feÄ^·«)"^^1·^1^)?^^)^).
A special case of this formula, when x = 0, is given in ([12]. p.212) and ([ 15], p.249). Since ψ (x) =0 , we find that
0 = αο(ψ^) + ( ΐ - ) α + 1 ( ΐ ^ ) Β + 1 ^ ( Α Α < α · « ) " ^ ^ 1 · Ρ + 1 > ( Α ) Ρ < α · « ( Α ) . k=i
Hence e(a,ß) (Ψ„.»*)
= - 0-*)α+1(ΐ+*)β+1 £ (fcÄ^^^^-^^ta:) P^'ß)(x) . k=n+i
We can further simplify this formula by observing that
/ Η ( α , β ) \ " ' = _ J _ Λ , « + ß + ' \ r(fe+1) Hfe+q+g-H) V fc / 2
α + β ^ 2 fe ^ T(fe+a+l) r(fe+ß+l)
and that, for as € (0,1) and a > — - , (3 > — - ,
p(a,ß)(a;) k
Μ(α,β) J_ ; ι - χ ) α / 2 + ι Α ( ι + χ ) 3 / 2 + ι Λ ' Λ
This inequality is a simple consequence of (7.32.6) in [14] . Using these results we find that
s(«.3)( ) = _ ( . - x ) a + 1 Q + x ) B + : y ρ(«+ι.β+ι>(Λ)Ρ(°.β>(χ) + 0(Λ n v rx y
9 a + 3 ^ fc-1 fc V71/ k=n+ l
and the estimate of the error term holds uniformly in [-1,1]. (4.4)
O r t h o g o n a l Po lynomia l E x p a n s i o n s of B o u n d e d Var ia t ion
If we w r i t e cos6 = x , we f ind t h a t
15
( i - * ) a + 1 ( . + * ) ß + V a ' ß ) ^ , * )
2 ( s i n 2a+2 ) 2 b+2 °°
V a ( a + 1 » 3 + 1 ) / a\ J a , 3 ) , Ωχ \ P ( c o s 6 ) p (cos6) L k-ι k
Using the asymptotic formula
/ . θ\<* + 1/2/ θ\3+ I/2 D(a,3)/ Qx I sin - 1 I cos - Γ P ,H/(cos9) =
we find that
θ \ 2 α + 2 / θ \ 2 β + 2 t a + i . e + i ) . I s i n — 1 I c o s y l P ( c o s 6 ) P (cos Θ)
2 a + l
) +°m 1 / / α + 3 + ΐ \ 2α+3 \ / Λ , α + 3 + 1 \ Ω 2α+ ΐ \ / ι \
- c o s i(2fe + α+3+1 ) θ - (α+1 ) π ) + o f - j ) · 2fc Hence
: ι - χ ) α + 1 ( « * * ) β + 1 δ β( α · β ) Φ τ . * )
_2α+Β+3 γ cos ((2fe + c t+ß+1)9 - ( α + ΐ ) τ τ ) η / ι ΤΓ- L
k=n+i °U)
Since 0 < θ < π , we obtain finally the estimate
n v Ύ x ' ,Μ(α,β;χ)
and Theorem 2 is completely proved.
16 R. Bojanic
REFERENCES 1. R. Bojanic, An estimate for the rate of convergence for Fourier series of
functions of bounded variation, Publ. Inst. Math. (Belgrade) 26 (40)(1979), 57-60.
2. R. Bojanic and F. Cheng, Estimates for the rate of approximation of functions of bounded variation by Hermite-Fejer polynomials, Canadian Mathematical Society Conference Proceedings 3(1983), 5-17.
3. R. Bojanic and Z. Divis, An estimate for the rate of convergence of the eigenfunction expansions of functions of bounded variation, Applicable Analysis (to appear).
4. R. Bojanic and S.M. Mazhar, An estimate of the rate of convergence of the Nörlund-Voronoi means of the Fourier series of functions of bounded variation, Approximation Theory III, Academic Press (1980), 243-248.
5. R. Bojanic and S,M. Mazhar, An estimate of the rate of convergence of Cesaro means of Fourier series of functions of bounded variation. This volume, pp.17-22,
6. R. Bojanic and M. Vuilleumier, On the rate of convergence of Fourier-Legendre series of functions of bounded variation, J. Approx. Theory, 31 (1981), 67-79.
7. R. Bojanic and D. Waterman, On the rate of convergence of Fourier series of functions of generalized bounded variation, Acad. Sei. Arts, Bosnia and Herzegovina, Radovi LXXIV (1983), 5-11.
8. F. Cheng, On the rate of convergence of Bernstein polynomials of functions of bounded variation, J. Approx. Theory 39 (1983), 259-274.
9. F. Cheng, On the rate of convergence of Szasz-Mirakyan operator for functions of bounded variation, J. Approx. Theory 40 (1984), 226-242.
10. Z. Divis, A note on the rate of convergence of Sturm-Liouville expansions, J. Approx. Theory (to appear)
11. G. Freud, Über die Konvergenz orthogonaler Polynomreihen, Acta. Math. Acad. Sei. Hung., 3 (1952), 89-98.
12. Higher Transcedental Functions, (Bateman Manuscript Project), Vol 2, McGraw Hill, 1953.
13. S.M. Mazhar, An estimate of the rate of convergence of the triangular matrix means of the Fourier series of functions of bounded variation, Collect. Math., 33 (1982), 187-193.
14. G. Szegö, Orthogonal Polynomials, American Math. Soc, New York, 1967. 15. F.G. Tricomi, Vorlesungen über Orthogonalreihen, 2nd Ed., Springer-Verlag,
New York, 1970.
AN ESTIMATE OF THE RATE OF CONVERGENCE OF CESARO MEANS OF
FOURIER SERIES OF FUNCTIONS OF BOUNDED VARIATION
R. BOJANIC* and S. M. MAZHAR**t *Department of Mathematics, The Ohio State University,
Columbus, Ohio 43210, USA **Department of Mathematics, Kuwait University,
P.O. Box 5969, Kuwait
1. Let / be a periodic function with period 2π and Lebesgue integrable
on [ -IT, π] . Let ^
2 a + / (a, cos kx + b-, sinkx) (1 .1 )
be the Fourier series of /. We denote by S (f9x) the partial sums of the Fourier series of /:
n
S (/>x)=2ao+ / (al· c o s ^x + ^l· s*n ) '
k=i
It is well known that if / is a function of bounded variation on [-ττ,π], then
lim (snf9x)-±f(x + 0) + f(x-0))) = 0 .
Recently the first author [l] obtained a sharper version of this result by showing that for n > 1
\snf,x)-\ (/(*+0) + /(*-0))|< \ \ V^ (gj . (1.2)
Here
x
| f(x + t) + fx-t) -f(x + 0) -f(x-Q) (t) = if t Φ 0 ,
[0 if t = 0,
and V Ag ) is the total variation of g on [0,t], t £ (Ο,ττ] . Later on the authors [2] showed that if p is a non-increasing
sequence of positive numbers and P = F. + . . . + P -*°° (n -*°°) , then for every function / of bounded variation on [-π,π]
fThe research of the second author was supported by Kuwait University Research Council Grant No. SM038.
MAA—C 17
18 R. Bojanic and S. M. Mazhar
Wnf,x) -%f(x + 0) + f(x-0)) l M k=0
Here W [f9x) is the [N 9p ) mean of (1.1) defined by
vn(f**)= f I Ρη.***(Λ*)
(1.3)
" k=0 From this result follow, in particular, the estimates for the Cesaro means of order a of the series (1.1) when aE(0,l] since in that case
n-l+ou / n -1 f α \
is non-increasing and, for α Φ -1 , - 2 , . .
Pn= I A£-*-A* £=o
n Γ(α+1) (n ->oo) .
The aim of this note is to obtain a generalization of (1.2) in a slightly different direction. We shall consider here the summability method for Fourier series of functions of bounded variation on [-ττ,ττ] defined by
Kn(f9x) = i J /(*+*) kn(t) dt (1.4)
for a general class of kernels k , which contains, in particular, Cesaro (C,a) means when a £ (-1,0], Clearly, we cannot expect here an estimate of the type (1.3) since in this case ^n~*0 (n-«>).
THEOREM 1 . Let kn(x) be an even, 27T-periodic and integrable function on [-π,π] satisfying the following conditions:
π M M t) dt = 1 ,
k (t)
o < An ,
and
*„(«> du (nt) l+a '
0 < t<π ,
0 < £ < π , a > -1
(1.5)
(1.6)
(1.7)
We have then the following estimate of the rate of approximation of / try Kn(f9x), the summability method defined by (1.4):
n Kjf,x) -j(/(* + 0) + /<*-(») .Af(ct)
1+ y *av"/k (9). ι+α U o x
k=i
R a t e of C o n v e r g e n c e of F u n c t i o n s of B o u n d e d Var ia t ion 19
For cont inuous f u n c t i o n s of bounded v a r i a t i o n , the p reced ing e s t i m a t e
becomes
Kn(f9x) -f(x) n l + a L χ-Tt/k
k=i As an application of Theorem 1, consider the (C, a) means σ (f9x) of
of the series (1.1) defined by π
%(/>*) =l\ fx+t) k*(t)dt, _π
where
n k = o n-k
and D (t) = (sin (n + |) t)/(2 sin til) is the Dirichlet kernel.
THEOREM 2. _If / €Β7[-π,π] , then for -1 < a < 0 , n
σ^(Λ*)-|(/<*+ο)+/(*-θ))|<^α Σ vavl/v(gx (1.8) v=x
For a = 0 we have from (1.8)
C(0) π/υ 1 > = 1
which gives the estimate (1.2). On the other hand if we assume that 2f(x) = f(x + 0) + f(x — 0) and that
4- O
V0^x^ = ° ( * ) f o r ° < t < π ' w h e r e 0 < 3 < 1 , α > 3 - 1, then (1 .8 ) y i e l d s
o^f.x)-fx)~o(n-a-1 £ v a " e ) = O(n^).
Thus we deduce the fo l lowing theorem of F l e t t [ 3 ] .
THEOREM A. Let / € Β 7 [ - π , π ] and l e t 0 < α < 1 , 0 < δ < π , α > 3 " 1 . I f x
i s a p o i n t such t h a t ^
b then σ " ( / , * ) - / ( « ) = ö ( n ~ B ) .
= ö(fcp) , 0 < * < δ ,
2. Proof of Theorem 1. In view of (1.4) and (1.5) we have π
Kn(f,x)-^(fix*o) +/(x-0)) -i J gx(t)kn(t)dt
Now writing
20 R. Bojanic a n d S. M. M a z h a r
ΤΓ τ\/η 7Τ
[ gx(t) kn(t) dt = J" gx(t)kn(t)dt + J gx(t)kn(t)dt π /η
we o b s e r v e , on u s ing (1 .6 ) t h a t π / η
\3XW-9X0) r I < ί
Il+I2, s ay ,
t
Andt < πΛ 70 g J < - ^ ^ v* VQ ( ^
V=l
Let A(t)
then
k (u) du ,
t π ΤΓ
^ 2 = J - ^ ( ^ ) ^ Λ ( ^ ) = - [ α ( t ) A ( t ) ] + f Λ ( * ) φ (*) π /η π /η
£)Λ(£) + j A(*)d^(t) π / η
Using c o n d i t i o n (1 .7 ) we have
i | ττ/η F 2
< 7 o <*
Jl
IT J/„ («*) (?„
π/η
= Γ 2 1 + Γ 2 2 · s a y .
I n t e g r a t i n g by p a r t s
*W*> π ^ .
and the l a s t term i s
22 1 + ot i 2 i + a + i J ^ α + ι ( η π ) n , t τ\/η
dt
π n ff/t ( a + l ) f l f
/ s a + 1 J ( η π ) j n - i t>+i
( a + i ) B V Γ ^α J ^
( η π )
Then
f =1 Ό
C2(a) « π / ΐ , < - = — - Y t> 7 (gr ) .
l + a / , o x v=l
, T i < B V > X ) C2(a) « / w
( η π ) η ί-1Λ
Λ ( α ) £ a π/ν ϊ Y υ ° Fo <**>
a;' (2.1)
(2.2)
Rate of Convergence of Functions of Bounded Variation Combining (2 .1 ) and (2 .2 ) we have
I g (t)k (t)dt\ < -K~- Y va I <*«>
This proves Theorem 1.
CAa) n ° 3 ν ^ V a ' π / f
(**>
M(a)
v = i n
ITS J ϋ ττ/f (^
V=\
21
Proof of Theorem 2. In order to obtain the rate of approximation of \f(x + 0) + f(x-O)) by 0a(f,x) for α € (-1 ,0] , in view of Theorem 1, we have to show that k (x) satisfies conditions (1.5), (1.6) and (1.7). Since
n c o n d i t i o n s (1 .5 ) and (1 .6 ) a r e c l e a r l y s a t i s f i e d wi th A = 2 , we have only to
v e r i f y t h a t c o n d i t i o n (1 .7) i s s a t i s f i e d when a € ( - 1 , 0 ] .
I t i s we l l known t h a t for a € ( - 1 , 1 ) we have ka(t) = R it) + S ( t ) ,
where ,
sin((n4 + f ) . - M R ( t ) = — nx ' a
n and
S (t)
[2 s i n (t/2)
2 α θ (t) n
Λ+α
n (2 s i n ( t / 2 ) )
where |θ ( t ) | < 1 (see [ 4 , p . 9 5 ] ) . Using the second mean v a l u e theorem and
the i n e q u a l i t y A > n v a l i d for a € ( - 1 , 0 ] we f ind t h a t
1 Rn^dv l + a
2<V + a «<
Next
l + a
2 (nt)
" „ i i π
r I 2 M f | S (v)dv\ < - i - L J nK ' | n ) t t
2 °° < π 1α1 f
<fl>
(2 s i n ( W 2 ) ) '
dv 2
*2\ a| 2nt 2
π l«L 2(nt) l + a
[ a € ( - 1 , 0 ] ) .
22 R. Bojanic and S. M. Mazhar From these inequalities follows that
\ ? i / l+a 2, | v fc (1;) du < I — — + — H - ) f+77
and condition (1.7) is satisfied.
REFERENCES
1. R. Bojanic, An estimate of the rate of convergence for Fourier series of functions of bounded variation, Publ. Inst. Math. (Beogrod) (N.S.) 26 (40) (1979), 57-60.
2. R. Bojanic and S.M. Mazhar, An estimate of the rate of convergence of the Norlund.-Voronoi means of the Fourier series of functions of bounded variation, Approx. Theory III, Academic Press (1980), 243-248.
3. T.M. Flett, On the degree of approximation to a function by the Cesäro means of its Fourier series, Quart. J. Math.(Oxford) 7 (1956), 81-95.
4. A. Zygmund, Trigonometric Series I, Cambridge Univ. Press (1959).
APPROXIMATION OF CONTINUOUS FUNCTIONS BY ULTRASPHERICAL SERIES
D. P. GUPTA* a n d S. M. MAZHAR** t
* Bright Star University of Technology, Brega. Libya **Department of Mathematics, Kuwait University, P.O. Box 5969,
Kuwait
1. The Jacobi polynomials P^ ' (x), α > -1, 3> -1, are defined by the generating function
2 α + 3 ( ΐ - 2 χ * + £ 2 ) " ^ [ ΐ - t + V ( l - 2 x t + t 2 ) r a [ l + t + V( 1 -2xt + t2)]~$ oo
0
The Jacobi polynomials P ' (x) are related to the ultraspherical (λ) η
polynomials P/J (x) by the relation „(λ-ί,λ-j), , Γ(2λ) Γ(η+\+|) ρ(λ) , . i en (χ> Γ(λ + |) Γ(η + 2λ) η l*' J λ > 2
and to the Legendre polynomials P„(x) by the relation
n x ' n
The ultraspherical polynomial, on the other hand,leads to trigonometric polynomial in the limiting case λ-*0, since
lim T p (cos Θ) = — cos ηθ , n > 1 . Λ η λ YL Yl
Let P (x) denote the orthonormalized set of ultraspherical poly-,(λ) n
nomials. From the orthogonality property it is seen that
(λ) , x ϊ (η + λ) Γ(λ) Γ(η+1) 1 ρν (χ) = \ *" L Γ(λ+£) Γ(|) Γ(η +
Ρ^λ) (x) , η = 0,1,2,.
2 λ- * Let (1 - £ ) 2ft) be Leb esgue integrable on [-1,1], then the Fourier-ultraspherical expansion of f(x) is given by
The research of the second author was supported by Kuwait University Research Council Grant No. SM038.
23
24
where
D. P. G u p t a a n d S. M. M a z h a r
0
1
= ( l - t 2 ) A ^ f(t)p™(t)dt
2. Suetin [l] considered the problem of approximation of f(x) by its Fourier-Legendre series. In this note we propose to examine a similar problem for Fourier-ultraspherical series. In what follows we prove the following theorem.
THEOREM. Lf / has p continuous derivatives on the interval ['- 1 , 1 ] , p
being a non-negative integer, then n
\f(*)- I ak?k (λ), = 0 1
k=o ( logn
+ 0\ -—r ω ,Ρ"λ 1 >λ > 0 ,
where ω(£) is the modulus of continuity of / " , such that t^ 2 iu(t) and A+p-\ ,,ν , t r ω(ί) are non-decreasing.
The above theorem includes as a special case for λ = * and ω(ϋ) = O(t^) , q>-y a theorem of Suetin. Possibility of this generalization was suggested by Suetin himself. He had also suggested that similar theorems can be set up also for Fourier-Jacobi series. However, so far we are unable to find such a generalization. 3. The following lemmas are pertinent in the proof of our theorem. LEMMA 1. (Timan [3], p.262). If the function / defined on [-1,1] has a pth (p > 0, an integer) continuous derivative, then there exists a constant M not depending on f9x and n such that for any n>p there is a polyno-
°° l· mial T (x) = \ byx of degree not higher than n which satisfies for every n o k
x € [- 1 , 1 ] the inequality
|f(,)-Tn(,)i<«p[ivC7 + Hp]Xa)[ivrr^ + M] , (3.,)
where ω(£) = ω(/ ; t) is the modulus of continuity of / * . LEMMA 2. (Szegö [3]. p.171). For_ 0< λ < 1 , 0 < θ < π
s m θ P!A )(COS8) "1 (3.2)
Approximation of Continuous Functions and
max -1 < # < 1
Ρ ( λ ) ( χ ) C2n λ > 0 ,
25
(3.3)
where Cj and <?2 a r e c o n s t a n t s .
4 . Prood of the theorem . We have
n n
f(x)- £ «kpiX)W\ = \f(*)-Tn(x) +Tn(x) - £ %P^\* k = 0 k=o
where T (x) is chosen to be a suitable polynomial of nth degree so as to enable us to apply the above lemma of Timan.
Expanding T (x) as a series of ultraspherical polynomials we have
71 fx)- £ akP(
kX)W
= | / (*)-rM(*) | + £ \0-t2)x-hTn(t)-fit))P(kx)it)p(
kX)ix)dt
k=o - l
I n
< | / ( * ) - v * ) | + ] ( ΐ - ί 2 ) λ _ ψ η ( ί ) - / ( ί ) | | £ ρ£λ)(ί)Ρ£λ)(*)
W 1
- 1
| f ( a : ) - T n ( a i ) | + ^ ( α ) , say
Applying Lemma 1, we have
1
k=o
( 4 . 1 )
tKP I_x)<M__ | " ' „ ~ + j ω
fc = 0
V i - t 2 | t | ^κ-2) 6?t (4.2)
Now for
and
\ « „2 / L « n2 J
2VTT-
p + i ( i - t 2 ) p / 2 f+/i-t2\ < 2 (4.3)
Similarly for
26 D. P. Gupta and S. M. Mazhar
— g < * < 1 , V ^ 7 * 2 < - ^
and
Λ l*IV ω Γ 2 I*
v r ^ 2 + 1*1
Combining (4 .3 ) and (4 .4 ) we have
' χ / Ϊ ^Τ 2 \t\\P (vLiiL + liLr \ n n2 )
V i - * % \t\ - n n2
< 2
< 2J
p+1
p+1 2p \n2 ) '
(4.4)
( , - t 2 ) P / 2 fs/T^ — ) n /
Thus
x I £ P £ X ) ( * ) P £ A ) ( * )
„ . p / 2
ί
dt
1 , 0 P / 2 , ι η
Ρ - * 2 ) fy/i-t2\(t +2\λ 2\ V ( λ ) ^ χ ( λ ) , ,
— Ρ — ω ν , — s — ; ν - * ; | 1 pi'(*>Pfc (*> η £:=ο
d t
+ Af λ " ' Ι V jVf^ju, dt
Ιλ + I2 , s ay . (4.5)
Using (3 .3 ) we have 1 n
:=o -1 k=o n
k=o
n ( 2 λ + 1 ' = 0 \ n
Applying Cauchy-Schwarz inequality
k=o
λ >0 .
Approximation of Continuous Functions
'.-°("-"Μ?)(ί ( ' - Γ ' Ι Σ -iu<'>-ί"<·> -l k=o
-1
(a;) is non-decreasing. Also
27
2d * ^
= 0 (4.6)
since χ^ 2 ω(χ) is non-decreasing. Also
1 P fc=o
dt
M (\ + f ) = Jll + J12 ' S a y : (4.7)
where Δ (x) i s the part on which | a ? - t | < 1 In and e (x) i s the r e s t of the
i n t e r v a l .
Now using (3 .2 ) and (3 .3 )
ν ( ρ + λ - 1 ) / 2 / v f i T £ 2 v , "
( λ ) / + Ν ^tt),„J dt) fe=o
η k=0
= θ(ηλ+1~Ρ | ω ( 1 / η ) d t ) = θ ( η λ ~ Ρ ω ( 1 / η ) ) , (4.8) V*> since ar ω(χ) is non-decreasing
use Ch
f>j»,(.)ri»W->i»M>ff,wt For the part e (x) , we use Christoffel-Darboux summation formula
l Ρ<λ>(*) Ρίλ ) ( . ) = e„ x-t
where Θ may be easily seen to be bounded as n-*°° . Since | t-x | > \/n
28 D. P. Gupta and S. M. Mazhar
for t € e (x) , u s ing Lemma 2, we f ind t h a t
!-£' x(p/2)+X-J/>/1-t2v
PiX)(t) p ( £ ) r w
-0(n^J ( V e (x) V
|*-t|
(ρ + λ-ι)/2 y — ^ d t
i t
V W . L r^ x-t 1
= 0 In P ω (i/w) log w J .
Collecting (3.1), (4.1), (4.5)-(4.9), we get the required result.
(4.9)
REFERENCES
1. P.K. Suetin, Representation of continuous and differentiable functions by Fourier series of Legendre polynomials, Soviet Math. Doklady 5 (1964), 1408-1410 (1965).
2. G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publications Vol. XXIII (1967).
3. A.F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press, New York, 1963.
MODIFIED SZASZ OPERATORS
H. S. KASANA, G. PRASAD, P. N. AGRAWAL and A. SAHAI
Department of Mathematics, University of Roorkee, Roorkee — 247667 (U.P.) India
1. Introduction. Papanicolau [4] obtained some results on Bernstein-type operators. In an analogous manner, Singh [5] defined a sequence of Sza*sz--type operators which maps the space of bounded continuous functions into itself as
oo
lSn9*W- I Vn,U(t)fX + ^)> ( K 1 )
k = o where -,
^η.κ^ ' - e — and x € [0,°o) k\
is fixed, and proved some approximation properties. In this paper, motivated by the recent work of Derriennic [1] on modi
fied Bernstein polynomials introduced by Durrmeyer [2] for functions inte-grable on [ 0, 1 ] , we propose a sequence of modified Szafsz operators defined on the space of integrable functions on [0,«>) as
oo oo
(Mn,xf)t) = Mn,x(fy);t) = n Σ Pn,lSt) \ Pn,ky) f(x+^)^> ( K 2 )
k = o ° where t,x€ [0,°°) and x is fixed. Clearly, (M f)t) is a linear positive operator.
Throughout this paper, C [0,°°) denotes the class of real valued» D
bounded and uniformly continuous functions on [0,oo) with norm U/H = sup \f(t)\.
B te[o,~) 2. Auxiliary Results. This section consists of certain results in the form of lemmas which we shall require to prove the main results of the paper.
29
30 H. S. Kasana, et al. For m€N (the set of non-negative integers), the mtn order moment of
the operator (1.1) is defined as CO
U t) = V V 7 (*) i--t) . n »777v ' L· n9kK ' \n J k=o
Consequent ly , U Q(t) = 1 and U ( t ) = 0 .
LEMMA 1. There ho lds t he r e c u r r e n c e r e l a t i o n
*V ^At) = t[U/ t) + mU At)] 9 m£N.
PROOF. It is easily observed that
tpn,kt) = k~nt)pn9kt) · ( 2 - ° Hence the result.
Thus,
(a) U ( t ) i s a polynomial in t of degree [m/2];
(b) For every t € [ 0 , » ) , U ( t ) = 0 ( n ~ [ ( w + 1 ) / 2 ^ ) , where [a] denotes
the i n t e g r a l part of a .
LEMMA 2. For_ m € N° , we have oo °°
fc=0 0 m / \ .
PtfOCF. We note that
! (fc+m)! k\
Therefore
y ( t ) - - i - f P ^ * ) 1 ^ · (2-3) Kn,777v ' 777 Z-. rn,kK ' k\ fc-0
For a l l t>y £ [Ο,οο) ,
JÜ. [tm e-ny-t)\ = V ^ - n j , (nt)k (k+m) ! ay-771 I J L ^! fe!
Applying Le ibni tz theorem in ( 2 . 4 ) , the l e f t hand s ide can be rewri t ten as
777
y Q^LtVe-"^--* ) (2.5) i) i\
i=0
Modified Szasz Operators 31
Replac ing y by t i n (2 .4 ) and ( 2 . 5 ) , we ge t 00 m
1 Ρη,,(ί)^7-= lUü* »· ( 2 · 6 )
Now, the proof is evident in view of (2.3) and (2.6). As a consequence of (2.2), we have
oo oo
n Ϊ pn,kw \pn9k^ (y-*)dy= ~ - * 4 · (2'7)
k = o 0 LEMMA 3 . Let the f u n c t i o n T (t) (πι£Ν°) be def ined as
oo ^
W ) = n I PnJ^\ Vn,k[y)[y-t)m dy> k=o o
then we have the fo l lowing r e c u r r e n c e r e l a t i o n
n,m+lK ' n,mK ' v ' n9rrv ' n , / w - l x '
PROOF. We have oo °°
fe=0 0 Using (2 .1) i n above t w i c e , we o b t a i n
°° °° t T n , J ^ = n Σ ^ - n t ^ n , k t ) \ P n , k y H y - t ) m d v - m t T n , m - l ^
k = o o oo
k = o ' o + nT _,_. t)-mt T At)
n ,m+1v ' n9m-lx ' oo oo
^ = ° ° + nT ( t ) - m i T . ( t ) OO OO OO
- « I ?«,*<*>! ρή k d / ) ( y - * ) m + 1 d » * t f p ; i f c ( » ) ( » - * ) m ^ k = o ° o
+ n T , ( £ ) - m t T (t)
--- ( m + l )T (t)-mtT ,(t) + nT _,_ t)-mtT At)
= - ( m + l ) r [t)-2mtT (t)+nT Α 1 (*) .
From t h i s the r e q u i r e d r e s u l t i s immediate . Now, (2 .8 ) t o g e t h e r wi th (2 .7)
32
imply
H. S. Kasana, et al.
(a) T (t) i s a polynomial in t of degree [m/2] ;
(b) For every t e [ 0 , « > ) , T ( t ) = 0 ( n~ [ (?7?+ * 7 2 ] ) ; and
(c) T ( t ) = - ( t + i ) .
n , 2 n n
LEMMA 4. There exist polynomials q . . (t) independent of n and /c such that
dtr n'k L V3 «i.j.rWPn,*(*>· 2^ + ^ r *£, J> 0
The proof of this lemma proceeds exactly on the lines of that of a result by Lorentz [3, p.26].
3. Main Results. In this section, we prove some approximation properties of the operators defined in (1.2).
THEOREM 1. Le£ f be integrable on [0,«>) and f£C [0,«). Then,
lMn9Xf)W-f(x + t) 1+2(t+l).f(^), where ω«(·) is the modulus of continuity of / on. [0,°o).
The proof, being trivial, is omitted.
THEOREM 2. Let / be integrable on [0,«) and f'£C [0,oo). Then,
Mn,xf)t) -f(x + t) n f
a. ~1/ 2 + n ω ..(.-"«> .κΓΐ.ψΗ)"1 where ω~/(·) is the modulus of continuity of f1 cm [0,°o).
PROOF. By the hypothesis
f(x+y) =f(x + t) + (2/-*)/'(*+*) + (3/-ί)/'(ξ) -/' (*+*) , where ξ lies between a; + z/ and x+t. Therefore, in view of (2.7) and Lemma 3(c)
OO GO
+ ^ Y Pn>fc(*) J p ^ t e ) (y-t)f/(i)-f/ (*+*) ^ . k=o o
(3.1)
Hence
Modified Szasz Operators 33
(MniXf)(t)-f(x+t) f «-> CX>
+ n Σ P n , k t ) \ P n , k ^ k=o
y-t f ' ( i ) - f (x+t) dy
f + Ix ( l e t ) . (3.2) Now,
J j < η ω „ / ("~1/2)Σ Pn,kW\ Pn,k(y)\y^\]+^T%dy= I2^I3 (say). k = Q
Using the Schwarz inequality for integral and then for summation
k=o
Χ 2 < η ΐ / 2ω / / ( n " l / 2 ) I P n J t ) ( \ Pn>ky) (y-t)> dy)
k = o o
<y(n-l/2)(n £ p^ fc( t) Ργι $ , (,)( y-t f dy) k = o o
= S — ( ^+7^) f ω ^ / ( η 1 / 2 ) ( i n view of Lemma 3 ( c ) )
Again, u s ing Lemma 3(c)
T I / 2
I3=n uy, <»-"2>!('4). Combining the estimates of I2 and J and taking into account (3.2), we get the required result.
THEOREM 3. Le_t / be bounded and integrable on [0,°o) and let f" exist at a point x + t £[0.°°). Then
lim n[M f )(*)- f(x + t)] = /' (x + t) +tf"(x+t) η,χ'
(3.3)
F u r t h e r , t h i s l i m i t ho lds uniformly i n x+t £ [ Ο , α ] if_
f"(x+t) £C[0,b), 0<a<b .
PROOF.. Since f" e x i s t s a t a p o i n t x + t € [0,<») t h e r e f o r e
fy)=f(x+t) + (2/ - (*+*)) f' (*+*)
where e(u)->0 as u->0 and is a bounded function on [-#-£_,<») . Set Af = sup I ε(^) |
w€ l-x-t ,00)
34 H. S. Kasana, et al
Hence, using (2.7) and Lemma 3(c), we have
n[(Mn>xf)(t)-f(x+t)]
where
= n[± f (x + t) + ±(t + ±) f" (x+t)\ + E(n,t) ,
OO °°
Ε(η>ϋ)=η2^ ? „ , & ( * ) Pn9k^ zy-t)y-t)2 dy . k = o 0
To prove the resu l t (3.3) i t is suff icient to show that E(n9t) -»0 as n->°° . Since e(y-t) -> 0 as y-*t, for a given ε >0 there exis ts a ό >0 such that
I e(y~t) I < e> whenever 0 < \y-t\ < 6 .
Let φΑ(ζ/) be the charac te r i s t i c function of y £ ( £ - 6 , £ + 6 ) . Then,
£ ( r z , i ) OO
k=0 OO
[ Ρ η ^ ( ί / ) | ε ( 2 / - £ ) | ( ΐ - Φ δ ( 2 / ) ) ( 2 / - t ) 2 ^ ]
= 1 ^ 1 2 , (say) . Evidently, in view of Lemma 3(b)
1,<εη2 £ Pn9kM j Pn9k(y)y-*)2dy
= ε·0(1), and
Γ < Mn2 V p 7 2 / , n , k
= 0n
k = 0 W vnyky)
[y-t)lq
2q_2 dy ; q (integer) > 2
Combining the estimate of II and J2 , due to the arbitrariness of ε > 0 , it follows that En,t) -> 0 for sufficiently large n,
To prove the uniformity assertion, it is sufficient to remark that δ in the above proof can be chosen independent of t £ [Ο,α-χ] and _Z\ and I2
tend to zero uniformly in t G [Ο,α-χ] for sufficiently large n.
THEOREM 4. Let / be bounded and integrable function on [0 ,<») admitting a derivative of order r at x + t £ ( 0 ,°°) . Then
t 3t (3.4)
Modified Szasz Operators PROOF. By the hypothesis
f(y) = Y l i-^±ll(y-x-t)i+e(y-x-t)(y-x-t)r ,
where e(w)-»0 as u->0 and is a bounded function on [-x-t,°°) . Set M = sup|e(w)|
w € [-x-t,00) Then, using Lemma 2, we have
35
— M f ) t ) 3f .3? η,χ'
£ *<A k = o 0 -z:=o
00
00 f
= f(r)(x + t) + n £ Ρ^( * ) J Vn%k(y) e(y-t)y-t)r dy . k = o 0
Hence for the existence of (3.4), it remains to show that 00 °°
n Σ Vn,k^ \ Pn,k^ Ζ^~^^~^Τ dy ° ' as 00* On an application of Lemma 4 and Schwarz inequality for summation, we get
I = tr(±- (M f)(t) - - £ - f(x + t)
n Σ Σ n' k- · (*) k-nt 3?n,kt)x
where
0
'—' 1 /2
x* ^ «*(y (fc-»*)2j,pn fc(t)) χ
00 00
( t)
k=0
2 \ l / 2
k=o
ΛΓ = max
i , j > 0 ^ , j , r '
Since ε ( 2 / - £ ) - * 0 as y^t, for a given ε > 0 t h e r e e x i s t s a 6 > 0 such t h a t
\e(y-t)\<e whenever 0 < -\y-t\ < 6 . For \y-t\ > 6 , the boundedness of
e(y-t) implies that
36 H. S. Kasana, et al.
\e(y-t)\ < Uy-t\' Thus
f Ί 2 o M2 9 | e ( z / - t ) j < ε 2 + - 2 - (y~ty fo r a l l z / € [ 0 , c o ) .
7 u s ing Schwarz i n e q u a l i t y for i n t e g r a l and ( 3 . 5 ) , we have oo 2 °°
( PnJy)e(y-t)(y-t)dy) < ^ J Pnik(y)^2 (y-t)lr + ^ (y-t)2r+2
(3.5)
o o Now, using Lemma 3(b) it follows that
2
Pn,kw\\ pn,k^ ^(y-t)(y-tf dy) 0
oo oo 1 V ,M\ f / \ J 2 , x ^ 2 r M , x v 2 r + 2 l
k = o
= ε20(η~(ρ+2)).
F i n a l l y , us ing Lemma 1(b)
I< Mln I 0(ni + U/2))zO(n-(r+2)/2) < M2 ε 2i + j <r
i , J > 0
Since ε>0 is arbitrary, J->0 as n->°° .
THEOREM 5. Le_t f be bounded and integrable on [0,«>) and
3 * ' / U + t ) e c [o ,b )
Then, for a l l s u f f i c i e n t l y l a r g e n t h e r e ho lds
sup d-KJ)^-h=f^^ dt 8 t x
< m a x c i W ( r ) ( ^ " 1 / 2 ) > C2n-(s~r)/2 ,
where C. = C,(r,x), Cn = C (r,x), 0 < a < < ? < £ , s >r> and ω , ( · ) denotes i l 2 j» ( P)
t he modulus of c o n t i n u i t y of / on [ 0 , b ) J
PROOF. By the h y p o t h e s i s , for x + y E [0 ,°°) and x + t E [α,θ]
/ ( * + ¥ ) - Σ f ( i ) ( . r t ( y - 0 * + ' ( r > < » - / ( l , ) < * + *> (3-6)
=y-t)r Φ(*+2/) + F ( x + z y , i ) ( l - c j ) ( x + 2 / ) ) ,
where $(x + y) denotes the characteristic function of x + y E [0, b) and ξ lies between x + y and x + t . Moreover,
Modified Szasz Operators 37
(ΐ) F(*+y,t) = / ( χ + 2 / ) - Υ f t
x + t ) (y-t)1 , t = 0
for a l l x + y € [0,<») and x+t € [a9c] .
Now, in view of (3 .6 ) we have
C» OO
k = o
+ -^- y ρ^(^) j ρη^(έ/)^(^+^^)(ι-Φ(^+^))Φ r\ L
k=o
= J i + J 2 + J 3 > ( s a y ) ·
From (2.2), we get
Using Lemma 4
Γ1 =^~ϊ /(*+*)· 3tr
Σ Σ »4 k-nt it-v
fc=0 2i + j < i > t , J > 0
1,3 3?
\fM(i)-fir(x + t)
(*) ?n,fc(t) J P„,i
y-t >(x+y) dy
. Mn < — ω , .
i , J > 0
"—t o
(i + ^ ) \ v - t \ ' dy.
^ + Ι 5 , (let), where
M = sup sup £ t € [a-x, c-x] 2i + j<3?
^ , J > 0
On an application of Schwarz inequality for summation and for integral and Lemma 1(b) and Lemma 2(b) we have
38
τ .Μηϊ,Ζ I. ^ ω
Η. S. Kasana, et al.
M<»-,/2> Σ .ft 2i + j < p
i , ΰ > ° k = o ?„,*<*>
. Mn 1/2 V
( vn^y) (y-t)2v dy) 0
OO oo
= o o
Σ n i + ^ ( n - i / 2 ) ö ( n - ( r + 1 ) / 2 )
^ , J > o
.. 1/2 , - 1 / 2 , Mn u> , .n )
fM
k = o
rl 2^ + j < p
i , J > 0
«y^n"1'2)*!). uniformly in x + t€ [a ,o]. By the same r e a s o n i n g , i t fo l lows t h a t
I , < ω , An~l/1) 0 (1)
uniformly in x + t €. [a , o] . Now, for x + yE [b ,°o) and x + t € [a 9 c] due to the boundedness of
[0,oo) and / r ) € C [ 0 , b ) we have v (η·\
y - t ) <κ \y-*\
where s>r and 6 is such that 0 <6 <b-c. Hence
q · · (t) oo
XMn 1/2 1/2
0
/c=o oo oo
2i + j <p 1/2
k = o 0n-(l3-s)/2)9
uniformly in x + t £ [a, e] . Combining these estimates of I —I we obtain the required resu
Modified Szasz Operators 39 THEOREMS. Let. / be in tegrab le on [ 0 , » ) . Then, l im (M f)(t) = f(x + t)
y%-*ca ΎΙ 9 X almost everywhere on [0 , oo) .
PROOF. Let y
F(x+y) = J f(x + u) du, 0
then F/(x + u)= f(x,u) almost everywhere in (0 ,»), Let a€(0,°°) where F ! (x+a) = f(x+a) . Therefore, to prove the theorem, it is sufficient to show that (M f)&) converges to f(x+a) as n->°o.
For all k, 0 <&<«>, the product p1 ,y)Fx + y) is absolutely continuous and because Ff(x+y) = f(x + y) almost everywhere we have for all t€ [0,°°):>
t t pn kt)F[x + t) =\ p'niky)F(x + y) dy+ J p^ (y) f(x+y)dy . (3.7)
0 0 ' Hence OO oo
( V * / ) ( a ) =~n Σ Pn,k^a) | V'n^y)^^y)dy. (3.8) The function F(x + y) being differentiable at x + a, we have
F(x+y) = F(x+a) + (y-a) F1 (x+a) + (y-a) e(y-a), (3.9)
where ε (η ) -»0 as η-*0 and ε(η) i s bounded on [-a, °°) . We put M = s u p | e ( n ) j .
η € [ - a , » ) Using (3 .9) in ( 3 . 8 ) , we get
(M f)a)= nF(x+a)e~na -nae~na F1(x+a) + F1(x+a)-n Yi 9 x
k=0 >w that
prove that
oo r~
L p n , k ^ \ ρ η , ^ ^ υ ~ α " e(y~a)dy
Now. to show that (M f)(a) converges to f(x+a) as ft-»°°, i t s u f f i c e s to
Rn(a) = n L P n t k ^ \ pn,k^^y~a^ ε ( ^ ~ α ) dy~* ° > a s η^°°-
k = o o
On u t i l i z i n g p'n k(y) = n (p n ^ k_ χ ( y ) - p ^ ^ k (y )) , we ge t
oo
Rna) = n2 γ (pn>fe+1(«) -P„>fc(«)) PM,fc^) (»-«) £(^-«) dl/
oo
aRja) = n ^ p ^ ^ fe + χ ( a ) ^ n a - ( k+ l )J J Vn^ky) ( 2 / - a ) ε ( ι / - α ) c?i/.
fc=o
Using Schwarz 's i n e q u a l i t y for summation, we have
40 H. S. Kasana, et al.
ΐ (αΕη(α)) < „ 2 ( Χ pn9k+i(a)(na-(k+i))2)
k=o o
Since e(y-a) -+ 0 , as y-*a, for a g iven ε > 0 t h e r e e x i s t s a ό > 0 such t h a t
I ε ( y ~ a ) I ^ ε whenever 0 < | y-a | < 6 . For | y-a \ > 6 , t he boundedness of
e(y-a) on [0 , ° ° ) imp l i e s | ε ( ζ / - α ) | < β/ "^Ä—^-. Again by us ing Schwarz 's
i n e q u a l i t y for i n t e g r a l
k = o
— ,' p k=Q
,k+i^a\ ^ ^ f e ) ( r « ) 2 * ( £ ( r a ) ] <*2/
n , H l ( a ) ί Ρ η , λ ( ί / ) ( ^ - * ) 2 ( * ( * " * ) ) di/ 1 V n L?
k=0 \y~a\ < δ
| l / - a | > 6
Γχ + J 2 , ( s a y ) .
Now,
fc=0 | z / - a | < 6 oo oo
n k=o o
= I ε 2 \ia 2 e - " a 1 η ε \~i a — J
ε ° v ^ and similarly
J 2 = n λ ρ η , ί = + 1 ( α ) \ Pn ,ky) y~a )Z \ £ y~a ]) &y
k=0 \y-a\>&
fc=o "o h lpn,k+ia) \pn,k^ (y~a)"dy
6Z ^nhJ
Modified Szasz Operators 41 Combining the above e s t i m a t e s of I and I , we ge t
Σ P n , f e + i ( a ) ( l p « , f e ( 2 / ) ( i / _ a ) z(y-a)dy) k=o
(''^)°(jr) - ε2 ί / ' ^
Now, s ince by Lemma 1
X P n , k ( t ) (k-nt) 2 = nt
and so
k lPn9k+ia)na~k+l)) is bounded uniformly in n. Hence, for sufficiently large n and ε>0, we have
aR (a) < Κε , nx '
where K is a constant independent of n and ε . Since ε>0 is arbitrary, R (a) -* 0 as n-*<*> .
This completes the proof.
Acknowledgements'. The authors are thankful to the referee for giving useful comments and Dr R.P. Sinha for helpful discussions.
REFERENCES
1. M.M. Derriennic, Sur V approximation de fonctions integrables sur [0,1] par des polynomes de Bernstein modifies, J. Approx. Theory 31 (1981), 325-343.
2. J.L. Durrmeyer, Une formula df inversion de la transformed de Laplace: Applications a.' £a theories des moments, These 3e cycle, Fac. des Sciences de £/ Universite de Paris , 1967.
3. G.G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953.
4. G.C. Papanicolau, Some Bernstein-type operators, Amer. Math. Monthly^ (1975), 674-677.
5. Suresh P. Singh and Om. P. Vershney, On Szasz type operators, Rend. Mat. 2 (1982), 565-571.
APPROXIMATION BY WALSH SERIES
A. H. SIDDIQI Department of Mathematics, Aligarh Muslim University,
Aligarh, India
Abstract. The investigation is related to some elegant properties of a Fourier series with respect to an interesting orthonormal system, known as the system of Walsh functions, which has found tremendous applications in the last decade. Two theorems are obtained, one of which deals with the degree of approximation by De £a Vallee Poussin means of Walsh Fourier series and the other is devoted to the study of a space defined with the aid of Walsh series.
1. Introduction. The trigonometric Fourier series has played a very significant role in the solution of various physical problems and its applications of vital importance have been found in telecommunication engineering and the analysis of stochastic problems. The natural problem arose whether one can find other orthonormal systems which can replace the role of the trigonometric system. Such an interesting orthonormal system was introduced by the American mathematician, J.L. Walsh, in 1923. A slight modification of this system was given by R.E.A.C. Paley in 1932 which is commonly known as the system of Walsh functions and the study of Fourier series with respect to this orthonormal system is known as Walsh-Fourier Analysis. Two celebrated research papers, one by Fine [5] and the other by Morgenthaler [11] provided firm foundation of mathematical properties. The study of the applications of Walsh Fourier series into telecommunication Engineering, initiated by H.F. Harmuth in 1968, showed that the Walsh system gives better results than the trigonometric system. More recently its application in electromagnetic radiation and radar have been investigated. The fact that Walsh functions take only two values +1 and -1 gives rise to favourable
43
44 A. H. Siddiqi repercussions in calculation and digital hardware. For a detailed account of mathematical theory developed upto 1972 we refer to Siddiqi [12, 13]. A systematic discussion on the applications of Walsh functions can be found in Harmuth [7,8], Beauchamp [3] and Maqusi [10]. For a complete bibliography of researches during 1971-81 and before 1971 we refer to Wade [14] and Balasov and Rubinstein [1] respectively. Several aspects of this beautiful system are yet to be explored. In this paper we have investigated the degree of approximation by De £a Vallee Poussin mean of Walsh Fourier series and have identified a space defined with the aid of Walsh series.
2. Definitions and Notations. The Rademacher functions are defined by φ0(χ) = 1, 0 < * < 1, φ0(χ) =-1, \ <x < 1, φ0(*+1) = Φ ο ω , Φη(χ) =φ02ηχ) , η= 1,2,3, ...
The Walsh functions are given by
Ψη<*) = ΦΜΙ(«) Φ„2(*) . . . Φ„γω where positive integers n . are uniquely deter-
The dyadic group G is the set of the sequences x = x consisting of 0's and l's, with termwise addition module 2, denoted by -f. The topology of G is defined by the system of neighbourhoods of identity
V = xeG, x. = ... x = 0 n L 1 n J
(with convenience VQ = G ) , or equivalently by the distance oo
X(x) = Y 2~n x . L n
tt=l The total measure of G is normalized to be equal to 1 . Walsh functions are identified with characters of G. A finite combination of Walsh functions,
n-i
ψ 0 (χ
for n = 2 L + 2 z + . . .
mined by n . < n . .
) = 1,
, + 2 γ
Σ v = o
c ψ (x) ,
is called Walsh polynomial of degree n if c _ ^ 0 . The set of Walsh polynomials with degree not greater than n is denoted by W . The Fourier series with respect to the Walsh system can be studied over [0 , 1 ] or G. It is a well established fact (see for details, Morgenthaler [11]) that Walsh Fourier series on [0,1] and G yield the same results for the classes of functions characterized by integrability conditions.
oo
1 an^nx)
n = 0 is called the Walsh Fourier series of a periodic integrable function fix) on G if
Approximation by Walsh Series
an = J fx) ψη(χ) dx .
45
n-i V sn(x) = sn(f,x)= I a^^x) = sn(f)
and
σ ix) = a if9x) n n J
^=o S0 ^ + Sl ^ + s2 (a?) + . . . + s ix)
are called the n-th partial sum and the arithmetic mean of Walsh Fourier
series of fix) respectively.
sn(x) + sn+l(x) + " · + s2n-l(x)
τ (x) = τ ( / , * ) = — — ^—^ n n J n
is called the De £a Vallee Poussin mean of Walsh Fourier series of fix) . A function fix) is said to belong to Lip iW) if
fix)-fix+h) sup xEG
E if) = inf sup
o(Xih))a\, a
fix)-Pix)\/pew ) 1/ n J
> 0
is called the best approximation of fix) by polynomials from W
We write
and
L =\f/\ \f\P dt exists! , 1 < p
/ (i / * dt y/p
A Walsh series / is said to belong to L if / is the Walsh Fourier c . o
series of some f£L . L is the collection of all Walsh series f which V P
satisfy | ( / ,/)|< °° for all ^° € Lq . 3. The Degree of Approximation by De la Vallee Poussin Mean of Walsh Fourier series THEOREM 3. 1. Let fix) be bounded and fix)£L1. Then
|τη(*)-/(*)| < 7 Enif) . COROLLARY 3.1.1. Le_t /(#) £ Lip°° (&0 , 0 < a < 1 , then
We require the following lemma in the proof of this theorem.
LEMMA 3.1. For any bounded function fix)
\τηχ) < 6 max x£G
fix)
46 A. H. Siddiqi PROOF. First of all we observe the following facts
For For fix) E L1 ,
fix)ewn, Tnif,x)= fix) (3.1)
J lim Tnif,x)dx= \fix)dx (3.2) Q n-»oo G
(3.1) is evident as Walsh polynomial is Walsh Fourier series of i tself .
and
kokix) = sQix) + . . . + sk_i(x)
ik+\) ok+lix) = sQix) + . . . + sk_xix) +skix)
By these two relations we get
ik+\ ) °k+1(x)- kokix) = skix) .
By writing k = n, n+ \ , n+ 2 , . . . , In - 1 and adding these relations we get
s ix) + s , Ax) + . . . + sn Ax) = In on ix) —n σ ix) n n+i 2n-r in n
τ ix) = 2 o0 ix) -oix) n tn n
(3.3)
By taking limit as n-χ», integrating over G and keeping in mind a well known theorem of Fine, which states that o9 (x) and σ ix) converge to fix)
almost everywhere, we get (3.2). By (3.3) we get
|τη(χ)| <2|σ2η(*)| + |σΜ(χ)| (3.4)
We know that (see Jastrebova [9] and Morgenthaler [11])
Okix) = | f(x + t) Kkit)dt ,
where G
k r-l
V«-(i) 1(1 V* r=\ v = o
By a theorem of Yano [16] and Theorem 1 of Margenthaler [11] we get j \Kkit) di < 2 , fc= 1,2,3,.... G
In view of the above results, we have
τ (a;) < 2 J /*(x + t) #2 (*0 <it
+ J /(# + £) K (t) d t
< 2 max \fix)\ \ \K (t)\ dt x G
Approximation by Walsh Series 47
+ max I/(a?) I [ \κ (£) I dt \Ί I J | n I
< 6 max I/(α;) | .
This proves the Lemma.
It is clear from the proof of the Lemma [see (3.3)] that τ (χ) con-n
verges to f(x) almost everywhere. By virtue of this lemma and the Lebesgue bounded convergence theorem
we get lim Γ τ fo) dx = J f(x)dx (3.5)
Proof of Theorem 3.1. We have 2n-l
xnf-g,x)-X- 1 8k(f-g,x)
k = n
2n-i in-i
k=n k=n or
Tn(f-g9x)=Tnf,x)-Tng,x) . (3.6)
Let w(x) €W be the polynomial of best approximation for f(x) . Then
\f(x)-w(x)\ <En(f) . (3.7)
By Lemma 1 , for g = w ,
\Tn(f-w,x) | < 6 max \f-w\ = 6 (f) . (3.8)
By (3.6) and (3.1) we get
and \Tn(f-w,x) I = \Tn(f,x) -Tn(w,x) I (3.9)
τ (w,x) = w(x) (3.10) respectively.
By (3.8), (3.9) and (3..10), we get
\Tn(f,x)-f(x)\ < 7En(f) .
This proves the theorenr .
Proof of Corollary 3.1.1 . For fE Liv°° (W) , 0 < a < 1 , E (f) = 0n~a)
by virtue of a theorem of Watari ([15],, p.4, case p = oo). Theorem 3.1 immediately gives the required result.
In view of the remark of Prof. P. Simon [see MR 84C: 42038] the above theorem can be easily established for f £ If , 1 <p <oo .
48 A. H. Siddiqi ö
4. I d e n t i f i c a t i o n of the space L K p
THEOREM 4 . 1 . For l < p < « > , - + - = 1 , we have
Le = L . p q
In the proof of this theorem we require the following lemma.
LEMMA 4.1. For 1 < p < » , - + —=* 1, the Walsh series CO
is the Walsh Fourier series of a function /€ L if and only if for every g € L with Walsh Fourier coefficients b, ,
oo
is (C , 1) summable
PROOF. Let oo
be the Walsh Fourier series of f€L , and let q € L and p y q
n-l V
sn^ = L °^^χ) '
In order to prove necessity it is enough to show that 1 1
lim I sn(x) gx) dx = \ f(x) g(x) dx (4.1) 0
n-l °khk'
o fc = o (4 .1) i s e q u i v a l e n t t o
1 l im [ g(x) f(x)-s (x)) dx = 0 . (4 .2 )
n-*oo 0
Since l im I I / - S || = 0 , 1 < p < oo by v i r t u e of a w e l l known theorem of n->oo n p
Paley ( see for example Wade [14]) we ge t (4 .2 ) in view of the r e l a t i o n : 1 f gx) (f(x)-s (x)) dx
1 0
Moreover, i
L Wf-Sn\\L · ( 4 ' 3 )
f \ \f(x)g(x) dx= L °]<
b]<' k=o
For the s u f f i c i e n c y p a r t , l e t g€ L and l e t the s e r i e s
Approximation by Walsh Series 49
i fe=o
°kbk
be (C, 1) summable. Let σ ix) be the (C,l)-mean of the Walsh series oo
I ckh(x)
k=o and let γ denote the (C, 1) mean of the series
Then
Σ e k h k -k=o
1 tt-1
\g(t)an(t)dt=\ g(t) £ ( 1 - JL-) ^ ( t ) <ft k=o
The relation γ ig) defines a linear functional on L . The norm of this functional is II σ II . Since the series
n p oo
L k k k = o
is (C,1) summable the values γ ig) are bounded for every g€L , that is,
J git)Onit)dt <Mig) <<*>,
where M is a constant, which depends on g but not on n and so
lim γ ig) < <» . By the Banach-Steinhaus theorem the norm of the functional γ ig) are bounded, that is,
< K for n = 1,2,3,.... (4.4)
Let F ix) = f σ (t) dt ,
Proceeding on the lines of Bary ([2], p.166) we can prove the existence of a subsequence Fn ix) of F ix) which converges to an absolutely continuous function F(x) . For each j yn . > i , we have
3 1
0 -4-) °i= i ψ<*>*ν<*>> where
^ = 2 ^ ^ - ^ / , 0 < ?:' < 2
50 A. H. Siddiqi Let
I. = 0
j/2 , j+1/2 , j=0,l,2, ... , 2 * + 1 - l ,
be the interval of constancy of ψ . (x) . Then
0-θ 2 ^ «* hKK1*1)·
Letting &->«> and replacing each F~ Or) by F(x) we get 0 1
e. = ί ψ .dF(x) .
Since F(x) is absolutely continuous Ff (x) = f (x) and so
-k (x) f(a?) dr. Since (C,a) mean (a > 0) of the Walsh Fourier series of an integrable
function f(x) converges almost everywhere to f(x) [see for example Siddiqi ([13], p.60)] χ
( \ |/|P dx) = ( \ \f\P dx) < inf ( J \on(x) \Pdx) < »
by Fatou's lemma and (4.4). Thus f£L .
Proof of Theorem 4.1. If / € L then by Lemma 4.1 / is the Walsh-Fourier series of an f^-L , i.e., / £L . Conversely, if f€ L and g € L then n_ 1
But l l f s n ( g ) . 0 1
^ = o
lim
as n-»<». It follows that II-1L H V ^ - i i i p - 0
(/.a-) = J /* is finite for all q E L , f£L .
This proves the theorem.
Approximation by Walsh Series 51 5. Conclusion. Theorem 3.1 is an analogue of a theorem of De la Vallee Poussin concerning trigonometric polynomials. Corollary 3.1.1 is an extension of a theorem of Yano concerning approximation by Cesaro means of Walsh Fourier series (see for example Theorem 2.1, Siddiqi [13]). Results in Section 3 are not exactly analogous to the corresponding results for the trigonometric system due to non-positiveness of the Fejer's kernel K-uit) for Walsh systems. Theoreom 4.1 can be quite useful for the study of the matrix transformation of Walsh Fourier series. Such problems for trigonometric systems have been studied in great detail by Goes (see for details, Goes [6] and his papers in the subsequent years).
The author would like to thank the referee for giving valuable suggestions for improvement in the presentation of this paper.
REFERENCES
1. L.A. Balasov and A.I. Rubinstein, Series with respect to the Walsh systems and their generalizations, J. Soviet Math, 1 (1973) 727-763.
2. N.K. Bary, A Treatise on Trigonometric series I, Pergamon Press, London, 1964.
3. K.G. Beauchamp, Walsh Functions and their Applications, Academic Press, New York, 1975.
4. De la Vallee Poussin J., Lecons sur I 'approximation des functions d'une variable reelle, Paris, 1919.
5. N.J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949) 372-414.
6. G. Goes, BK-Raume und Matrix transformation für Fourierkoeffizienten, Math. Zeit. 70(1959), 345-371.
7. H.F. Harmuth, Transmission of Information by Orthogonal Functions, Springer Verlag, Berlin, 1972.
8. H.F. Harmuth, Sequence theory — Foundations and Applications, Academic Press, New York, 1977.
9. M.A. Jastrebova, On the approximation of functions satisfying a Lipschitz condition by the arithmetic means of their Walsh Fourier series, Amer. Math. Soc. Trans. Ser. 2, 77(1968), 1966, 149-162.
10. M. Maqusi, Applied Walsh Analysis, Heyden and Sons Ltd., London, 1981. 11. G. Morgenthaler, Walsh Fourier series, Trans. Amer. Math. Soc.
84 (1957) 472-507. 12. A.H. Siddiqi, Walsh Fourier Reihe Arbeitgemeinschaft Von Prof. Leptin,
Heidelberg University, 1971/72. 13. A.H. Siddiqi, Walsh Functions, AMU Press, Aligarh, 1977. 14. W.R. Wade, Recent developments in the theory of Walsh series, Internat.
J. Math, and Math. Sei. 5 (1982), 625-673. 15. C. Watari, Best approximation by Walsh polynomials, Tohoku Math. J.
15(1963), 1-5. 16. S.Yano, On the Walsh Fourier series, Tohoku Math. J. 3 (1951), 223-242.
ON SOME CLASSES OF BI-UNIVALENT FUNCTIONS
D. A. BRANNAN* and T. S. TAHA** * Faculty of Mathematics, The Open University, Milton Keynes, UK
**Department of Mathematics, Kuwait University, Kuwait
1. Introduction. In th is note we discuss several classes of functions
oo
f(z) = z + £ anzn , (1.1) n-i
that are analytic and univalent in the unit disc U = z : |z|<l. The class of all such functions we denote by S. We denote by σ the class of all functions of the form (1.1) that are analytic and bi-univalent in the unit disc, that is f£S and / has a univalent analytic continuation to |u|<l. We also introduce the following classes:
(i) The class S [a] of strongly bi-starlike functions of order a, 0<a<1.
(ii) The class S (3) of bi-starlike functions of order 3, 0 < 3 < 1.
(iii) The class C (3) of bi-convex functions of order 3, 0<β<1.
For the above classes we give bounds for \a | , \aA\ also for the class C (0) we give the bound for \a I and the extremal function.
The class σ was first investigated by Lewin [1]; he showed that \a | < 1.51. Later Brannan [2, Problem 6.82] conjectured that \aA < /2 . The class S [a] and the class C (0) =C were first introduced in [3]. σ σ σ
2. The Class S*[a].
A function f(z) of the form (1.1) belongs to the class S [a], 0<a<l, if it satisfies the following conditions:
/ € σ , (2.1)
53
54 D. A. Brannan and T. S. Taha
arg
a r g -
zf ( a )
wg1 (w)
air
g(w)
\z\ < 1 ,
M < i ,
where j(w) = w — a0w + (2a9 -a~) w3 + . . .
-1
(2.2)
(2.3)
(2.4)
i s the e x t e n s i o n of / to the whole of | ω | < 1 .
THEOREM 2 . 1 . Let c»
fz) =z + ^ anzn ,
belong to S*[a], Then 2a
n = 2
and \a3\ < 2a /l +a
PROOF. We are going to follow the notation used in [4]; namely, we denote by P , 0<α<1 , the class of functions
J a oo \~ k
P(z) = 1 + > p-z , fc=l
that are analytic in the unit disc U and subordinate to the function [(1 +z)/(l -z)]a. Now, P(2)EP a if and only if P(s) = [^(s) ] α , where M ^ ) € P, ; and P is the class of functions of positive real part in U .
Conditions (2.2) and (2.3) can be written as
and (2.5)
(2.6)
respectively, where ζΚ^) , P(w) belong to P, and have the forms
and $(2) = \ + c z +o z + . . . p(w) = 1 +piW + p2W2 + . . .
If /(a) €5*[a], then by (2.5) zf* ( ) Γ^/ \ ΊθΙ Γ , , l . 1^
From this, it follows that
Also by (2 .6 )
2 , , α ( a - 1 ) 2 2aQ = a 0 + a c . + — c\ 3 2 2 o 1
^ = [p(u)f=[] + PiW+P2W* + ...f
S o m e C l a s s e s of B i -un iva len t F u n c t i o n s 55
This g ives an = — ap
o 2 0 , . a ( a - 1 ) 2 3 a 2 = 2 a 3 + a p 2 + _ ! _ L p^ .
Combining the s e t of e q u a t i o n s for α^,α^ we o b t a i n
2 <*2K+P2) α0 :
a + 1 By a well known theorem due to Caratheodory [5, page 41], \p | < 2,
\cn\ < 2. Hence I I 2q
For a we have 4a3 = ap2+3c2) +2a(a-\) β2
χ . (2.7)
If a = 1 , then | a J < 2 . So we consider the case 0 < a < 1 . By (2.7) 2
4Rea3 = a Re p2 4- 3c2-2(l -a) βγ . (2.8)
For the functions $(2) , P(w) , Herglotz's representation formula [5, page 40] states that
2π
«(2>= 1 ' + 2 e - ^ t * Μ * > · aIld 27T --t
o ' _ w e
where μ . ( t ) a r e i n c r e a s i n g on [ 0 , 2 π ] and μ . ( 2 π ) - μ . ( 0 ) = 1, i - 1 , 2 .
We a l s o have 2π
σ = 2 [ e"™* d p , ( t ) , n = 1 , 2 , . . . , n j 0
and 2π
Pn = 2J e~int dM2(t) , n = l , 2 0
Now (2 .8 ) becomes
2 π 2TT
4 R e a 3 = 2a cos It d\x(t) + 6a cos It d\il(t) 0 ° 2π 2 2π 2
- 8 a ( l - a ) j ! cos £ d y x ( t ) - | s i n t dv1(t)\ \ ,
56 2π
D. A. Brannan and T. S. Tana 2ir 2π
< 2a cos It d]i2(t) + 6a cos It d]\ ( t) + 8a(l - a ) sin t d]i1 it)
= 2
u o 2TT 2 π
a j l - 2 s in 2 tdy ( £ ) + 3 - 6 s i n ^ d y ^ t )
+ 4(1 - a ) j I s i n t d\\ ^t) 1 j .
By Jensen*s inequality [6, page 61], we have that 2π
j | s i n * | <2μ ( t )
Hence 2π
2π
< sin2£ ciy(t) . 0
2π
4Rea < 2 a < 4 - 2 j s in 2 t d\i2(t) - 2(1 + 2a) s i n 2 t ^μ (t) L
o o Therefore Re a < 2 a, which implies
| a 3 | < 2 a .
The effect of the bi-univalency condition can easily be seen by looking at the coefficients of the corresponding class S [a] introduced in [4]; this is the class of functions / of the form (1.1) univalent in \z\ < 1 and satisfying the condition (2.2). The sharp coefficient bounds are
and if
if
and if
0 < a < ±
i < a < l
a = 3 >
\a2\ < 2a
then ja j < a ,
then |a3 | < 3 a2 ,
then |a3| <\-
in each case the stated coefficient bound is sharp. It would be of interest to know what the sharp bounds of the coeffi
cients a2 , α3 are in the class S*[a] .
3. The Class £*($) . We define the class S*(3), 0 < 3 < 1 to be the class of all functions of the form (1.1) satisfying the following conditions:
feo,
Re
Re
If (a) J (wer' (w) \ 1 g(w) J
\z\ < 1 ,
\w\ < l ,
(3.1)
(3.2)
where gw) is the same function as in (2.4). We call S (3) the class of bi-starlike functions of order 3 ·
Some Classes of Bi-univalent Functions 57
THEOREM 3 . 1 . .Let, CO
f(z) = z + Y α^ zn
belong to S*(3) , (0 < 3 < O . Then
\a2\ < v"2(l-3) .and |a3| < 2(1 -3) .
PROOF. Let P(3) be the class of functions 7(s) analytic in 131 < 1 with 7(0) = 1 , Re V(z) >3 in |z| < 1.
In fact P(0) is just the class of functions
POO = 1 + ρχζ + p2z2 + ... , for which Re P(z) > 0 .
Note that V(z) E P(3) if and only if
P(z) = — — (7(2)-3) belongs to P(0) . l - 3
Hence, it follows that there exists a unique P(z) E P(0) such that
7(2) = 3 + (i_3)p(z) , (3.3) for 7(s) in P(3) .
Now conditions (3.1) and (3.2) are equivalent to Zff$ = 3 +(1-3) Q(z) , (3.4)
and ^l!M_ = S + (l-3)P(w) , (3.5)
respectively, where Q(z) , P(w) belong to P(0) and have the forms
2 ρ(2) = i + βλζ + e 22 + ... ,
and
P(JS) = 1 + pxw + p2w2 + . .. .
Now, it follows from (3.4) that
a2 = (1-3) ολ , (3.6) and 2a3 = a2 0 _ 3 ) öi + °~3) °2 ' ( 3 , 7 )
Also from (3.5) it follows that a2 = - ( 1 - 3 ) P l , (3.8)
and ka\ = 2a3-a2 ( 1 - 3 ) ρχ + ( l - 3 ) p 2 · (3.9)
The four equations give la1 = (1-3)(C +p ) .
Using the bounds for \e | and \p \ , we obtain
58 D. A. B r a n n a n a n d T. S. T a h a
and
\a2\ < \ / 2 ( l - 3 ) ,
\a3\ < 2 ( 1 - 3 ) .
In comparison, let S (3) , 0 < 3 < 1 denote the class of functions star-like of order 3 in |s| < 1; this is the class of functions / of the form (1.1) univalent in \z\ < 1 and satisfying the condition (3.1). It was shown in [7] that the sharp coefficient bounds for a2 , α3 are
|α2Ι < 2(1 -β) ,
\a3\ < (l-B)(3-2ß) . It would be of interest to know what are the sharp bounds for the coeffi
cients a2 , α3 in the class S (3) .
4. The Class C (3) . A function f(z) of the form (1.1) belongs to the class Ca(3) if it satisfies the following set of conditions:
/ € σ , (4.1)
R e ^ S l · ] > β * M <] (4,2)
R e W ^ ) + i> 3 , \w\<\ 9 (4.3) g (w) J
where g(w) i s the f u n c t i o n def ined i n ( 2 . 4 ) .
THEOREM 4 . 1 . Let oo
f(z) = z + ^ an zn ,
belong to C (3) . Then n=2
\a2\ < V T - 3 and \a^\ < 1 - 3 .
Mor eover, for the class 0(0) , the extremal function is given by
and its rotations.
PROOF. Using the same notation as in Theorem 3.1, conditions (4.2), (4.3) give
f (2) and // / N
^ T T * 1 = ß + (l-ß)P(w), (4.5)
where £(2) , P(w) € P(0) .
Some Classes of Bi-univalent Functions 59 Equation (4.4) gives us that
2a2 - (1-3) ολ (4.6) and
6a3 = (1-3) c2 + 2a2(l-ß)ö l . (4.7)
From these two equations we obtain
6a3 = 4a* 4- (1 -3) 2 . (4.8)
Now, by (4.5) we obtain that
2a2 =-(1-3) ?i (4.9) and
12α* = 6α3+(ΐ-3)ρ2-2α2(1-3)ρι . (4.10)
The two equa t i ons g ive
8a22 = 6 a 3 + 0 - ß ) p 2 . (4 .11)
Combining (4.8) and (4.11) and using the bounds for \p | and \o | , we obtain that
| a J < \ / T = ~ 3 and | α 3 | < 1 - β .
In the case 3 = 0 , we have C ( 0 ) c C , where C i s the c l a s s of a l l
normal ized func t ions convex i n the u n i t d i s c . This i m p l i e s t h a t
\a I < 1 , n = 2 , 3 , . . . ,
which is sharp as seen from the function
/<2> = τ4τ3 . (*·»2) which is in C (0) .
The question arises whether the class C (0) and the class C are the same. The function
belongs to C; since it is not bi-univalent, it is not in C (0) , consequently C (0) is a proper subclass of C .
We emphasize that, it is not true that: A function f(z) is bi-convex in U if and only if zf'z) is bi-starlike in U. This is clear from the function in (4.12) which is bi-convex, however for that function zf'(z) is the Koebe function which is not bi-starlike (since it is not bi-univalent).
D. A. Brannan and T. S. Taha
REFERENCES
M. Lewin, On a c o e f f i c i e n t problem for b i - u n i v a l e n t func t ions , Proo. Amer. Math. Soc. 18 (1967) , 63 -68 . D.A. Brannan and J.G. Clunie, Aspects of Contemporary Complex Analysis, Academic Press, 1980. T.S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981. D.A. Brannan, J. Clunie and W.E. Kirwan, Coefficient estimates for a class of star-like functions, Can. J. Math. 22 (1970), 476-485. C. Pommerenke, Univalent Functions, Vandenhoek and Ruprecht, Gottingen 1975. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York 1966. M.S. Robertson, On the theory of univalent functions, Ann. of Math. 37 (1936), 374-408.
SOME APPLICATIONS OF GENERALIZED CALCULUS TO DIFFERENTIAL AND
INTEGRO DIFFERENTIAL EQUATIONS*
M. A. AL-BASSAM Department of Mathematics, Kuwait University, Kuwait
1. Introduction» The title of this paper may suggest that the subject under discussion deals with the differential or integro-differential or integral equations of fractional (generalized) order. This is not the case, but in previous work [4], [6], [9], [13], some studies have been made by this writer, in addition to some other authors who have discussed some aspects of this topic [14]. Recently K. Nishimoto and others have made some studies about the integro-differential equations of fractional orders in the complex domain [18], [19].
In this article a study will be made about the use of Fractional (Generalized) Calculus methods in solving some classes of differential and integro--differential equations. This may be accomplished through representations of equations by their equivalent operator equations or, as they are called in some other papers, transform equations. Then the operator equations may be solved by using some operational properties of the integro-differential operators of fractional (generalized) orders (see [l]). Some of this work have been presented in detail by this author, (see [2], [3], [5], [7], [8], [10], [11]). Also, Mambriani's work, [15] and [16], may be considered as being based on related ideas.
So, this article contains two main parts of the subject: the first is dealing with equivalence relations and properties and the second part offers some methods of solving the operator equations which yield solutions to their equivalent differential or integro-differential or integral equations. In
Invited talk.
61
62 M. A. Al-Bassam particular the method will be applied to classes of differential or integro--differential equations whose solutions represent some generalized special functions such as hypergeometric functions, Laguerre functions, Legendre functions, Bessel functions, etc.
It may be pointed out that the "H-R transform equations" presented by the author in previous work will be called henceforth "operator equations". If some mathematical symbols or notations are not clearly defined they are found in other stated references of the work of the author.
1. SOME EQUIVALENCE PROPERTIES
1.1 Preliminaries and Definitions. An operator equation is a functional equation which contains the operators of generalized order of an unknown function, given in the equation, which is assumed to be defined and satisfy certain conditions over an interval (a, b). It may be represented by a differential equation or an integro-differential equation or an integral equation. For example the operator equation:
X /X ηΠ \ I px) y(x) = f(x) , ! J =D = — - i ( K ] ) Ct Ct CLJL
where n is a positive integer > 1 , p , y € C and f G C on a < x < & is the differential equation
n
L \k) V y = ^ k='o
where (n\_ Γ(η+1) V^y T(n-k+\)T(k+\)'
The operator equation x
D y + p i (x) Dy + p^ (x) y + p^ (x) i a y
where x x / ? a > 0 , Iay = - i - [ (x-t)a λ y(t) dt , n Γ(α) J a
is an i n t e g r o - d i f f e r e n t i a l e q u a t i o n . Also the o p e r a t o r equa t ion
Χλ-Γ ( 1 - α ) J 1 au(x) = fix) (1 .3 )
a where 0 < a < 1 , f(x) £C on [a,b] and u(x) is the unknown continuous function represents the Abel's integral equation.
The operator equation may assume some other forms different from the ones already mentioned which appear in the forms of successive operations or in the forms of composition of functions such as
Some Applications of Generalized Calculus 63
I~WP(x) I~lP(x) IW~n+ly(x) = fix) a l a 2 tt
where P. and P are given functions of the variable x and possess as many ΎΙ X
derivatives as it is required, y £ C and f€C in the interval a< ,<b.
This operator equation may be represented by a differential equation or an integro-differential equation or an integral equation. Such representation, however, depends upon the function forms of P. and P , the values of W and the positive integer n. These types of equations may be called "equations of successive operators".
This work will be confined to the study of some aspects of certain types of operator equations of successive operators and their equivalent equations that may be obtained by applying some operational properties of the operator of generalized order which are in essence the same properties as those possessed by the integral or derivative operator of generalized (fractional) order [1].
Definition. A differential equation or an integro-differential equation is said to be equivalent to an operator equation of successive operators if the first is an expanded form of the second.
For obtaining the expanded forms, the generalized Leibnitz rules of differentiation (see [1], pp.5-9) are used in the expansion process. The main formula of Leibnitz* generalized rule is given by
I f = L \i)f <*> I * a i = 0
and
p=0
where fix) i s a polynomial of degree n , R e a > 0 , R e 3 > 0 and
(&\= Γ ( β + 1 ) \PJ~ Γ ( 3 - ρ + 1 ) Γ ( ρ + 1 )
According to this definition (1.1) is equivalent to (1.2) and the operator equation
I~W e~fx) I~l e^x) IW eVx) y(x) = 0 , (1.4) a a a
where w is a number, , fix) = — x ,
0
64 M. A. Al-Bassam
x Σ s . t
y J n . ' t τ αη τ\* a Σ a . t M - ^ °
i -0 * a,,b, ( k = 0 , l , 2 , . . . , n ) are numbers and a # 0, is equivalent to the linear differential equation of the f i rs t order
The equivalence can be easily established by expanding (1.4) and writing i t
in the form
ΧΓΌ e~f(x) \f (x) IW eg(x) y + e^x) I » " 1 e^x) y o ,
which gives equation (1.4) in the form
y' +[f' (x) + g1 (x)]y = 0 .
1.2 Some Classes of Equations of Successive Operators. (1) Let w, k:i a. and b. (ϊ = 1>2, ... , m) be numbers such that not
all the b .' s are zeros and Rew>0. Suppose y € C on the interval a yb)
and m9 n are positive integers. Then the operator equation
x m / \ \ - k . x m / \k. x
represents, for m= n, a linear differential equation of order n and of the form
(E,), TT (ai + bix)Dny+ y Pn_r(x)Dn ry = 0,
where D = d/dx, D°y=y and P _ (x), for fixed a?, is a polynomial of degree n n r
less than the degree of Π (a . + b . x) by 2°.
If the transformation yx) = e Y(x) is applied to (E ) then the
operator equation
A-k. x m / \k. x (E2) i"» TT fc + b. x) * I~l TT (a. + J J l ^ + X *"μ* *(*> = 0 , for m = n , is equivalent to the ntn order differential equation of the form
n n
(EJ, TT a. + b.x)DnY + 7 P (αθ P"2"21 Y = 0, 2 1 . ' ^ ^ L· v p=l
Some Applications of Generalized Calculus 65
where P Or) , (r = 1 , 2 , . . . , n) are polynomials of at most ntn degree, i.e. r n they are of the same degree as that of the polynomial Π (a . +b . x) . The
i-\ ϊ I* forms of these polynomials can be easily determined.
It may be pointed out that the operator equations (E ) and (E ) may no longer represent linear differential equations when m >n , (for details see [2J and [3J.
It is interesting to note that if the a.'s as well as the 2? .! s are y ^ n
all equal, i.e., a . = A and b . = B for all (t=l,2,...,n) and k = Σ k. , then the operator equation
(E ) I (A+Bx) I (A + Bx) I y(x) = 0, a a a
is equivalent to the Euler linear differential equation
(E3)l (A + * r ) V „ + £ [P /(- n ) r -,(^ i)(- M + 1 ) r_ 1] 2 » = 1
X (-l)V(u + Ba:)n-I,D''-rJ/ = 0,
where (a) =a(a + 1) ... (a+r- 1), ( a)=l,a=£0. The equivalence of (E„) and (E ) can be shown easily by using the
properties of the differential operator of generalized order, as equation (E~) can be written in the form
X X X X Sfc I (Λ + 5χ) J z/ + J (i4 + 5#) J y = 0, a a a a
which, after expanding the terms, becomes
+ i("lf Q-n)rlf(A+Bx)n-*Dn-vy =0.
This equation takes the form of (E~), by replacing r, in the first summation, by r- 1 .
The significance of the equivalence between operator equations and their corresponding linear differential equations or integro-differential or integral equations is that once the equation is transformed into an equation of successive operators, the problem of finding solutions of an equation becomes easier to deal with through the use of some operational properties of the integro-differential operator of generalized (fractional) order. This matter will be clarified in the next sections when the problem of solving certain classes of operator equations is studied. MAA—F
66 M. A. Al-Bassam (2) For other cases of (E ) we may present the following: Let a ., a . be numbers, R (λ-w) > 0 (λ= 1,2,...), fix) is a given
τ ^ (0) in) function of Class C , yix) is a function of Class C on [a,b] and n and n are positive integers such that
1-a. x m a. x_ m / \ ^ x (E.) Γυ Τ]χ-α.)τ Γ1 Tjlx-a.) JW"n+1 y(x) = fix) .
t = l ^ = l a
(i) If rn = n, then (E ) is a non-homogeneous differential equation
By using the differentiation properties of the operator of generalized order it can be easily verified that equation (E,) is equivalent to the integro-differential equation p+2
m-2 w- Σ a . + 1 (Vi T^+ 1 φ
Ρ >, ^ X P = -l Κ^<ΐ2<...<ίρ+2</π ^ (x-ak)
xlp~n + 2 yix) = fix) a
where φ_ = 1 , φ = w(w- 1) . . . (w - r), (p = 0 , 1 , . . . , m - 2) . From the equivalence of these two forms, we may conclude the following:
( of n order of the Fuchsian class, with singularities at x = a. (-£=1,2, ...,n) .
(ii) If m>n im-n = v), then (E ) is a non-homogeneous (n order differential —V order integral), integro-differential equation of Fuchs--Volterra type with V integral terms containing integrals of the operator
xQ or integral orders of the forms IHy, iq = l,2,...,v). The equation's a
singularities are at a ., ii = 1 , 2 , . . . , m). (iii) If m < n (n-m = μ) then (E ) is a non-homogeneous differen
tial equation of n1-" order with the last term containing μ dirivative of y.
(iv) For the particular case m = n+ 1 , equation (E,). is a non-homogeneous, Funchsian type n order differential equation if and only if
k = n+i w - V a . + 1 = 0 .
L ^k k=i
Details about equivalence and particular cases may be seen in [2], [3J. 1.3 Some Cases of Equivalence Properties
Property 1. The necessary and sufficient condition that the second order linear differential equation
ia + bx) ic + dx)Yn + (αχ + b γχ) Yf 4- d Y = 0 , (3 .1)
i s e q u i v a l e n t to t he o p e r a t o r equa t ion
Some Applications of Generalized Calculus 67
I~w (a + bx)1~p(o + dx)1~q I~l(a + bx)p (a + dx)q J w _ 1 y = 0 , X° X° *° (3.2)
where a ,b 9o ,d , a ] , b , d , W,p and q are numbers such t h a t a and c a r e not
both zero and bd=^0, i s t h a t
bcU + adV = a1 (3 .3 )
bd(U + V) = bY ( 3 .4 )
bdw(U + 7 - ω - 1 ) = dj (3 .5 )
where U = W + p and 7 = w + ς .
I t may be po in ted out t h a t the v a l u e s of w , i n t h i s equ iva l ence case
a r e the r o o t s of the q u a d r a t i c e q u a t i o n
bdwz + (bd - b ) w + d 0.
Property 2. If the differential equation (3.1) is equivalent to the operator equation (3.2), then the necessary and sufficient condition that the equation
PQte) y" + P 1 ( x ) z / / +P2U) y = 0 , (3 .6 ) where
PQ(x) = (α + kc) ( e + a r )
P (x) = 4 x 2 + B x + C 1 1 1 1
P2(x) = A2x2+B2x+C2
and A , 5 , C , ^ , 5 , C a r e c o n s t a n t s , i s e q u i v a l e n t to the o p e r a t o r
equa t ion (3 .2 ) wi th Y = (exp \ix)y , \ι Φ 0 , i s t h a t :
bd\i2+2\i) = Αγ +A2 (3 .7 )
Bl = bl + 2 μ 5 ( 3 ' 8 )
C = a -2\xac (3 .9 )
^1 = ^ " ^ l ^ 1 + a c y 2 (3 .10) and
;T = 4&di42 (3 .11)
Βγ = \iS + B2\Tl (3 .12)
where S = be + ad.
Details of these properties are found in [3],
Property 3. (A) The necessary and sufficient condition that the differential equation
(a + bx)Y" + (a + b x) Y1 + d Y = 0 (3.13)
68 M. A. Al-Bassam
is equivalent to the operator equation
I~aa + bx)l~k eVX j " 1 ( a + bx) k e~VX J a _ 1 7 = 0 , (3.14) xo xo x
0
where a,b,a , b , d , a , k and μ a re numbers such t h a t b^Q, b d Φθ, i s
t h a t
a = — — (3.15)
(3 .16)
k = — , (3 .17) (7<*2>
f(s) = as2 + a.s + d, , g(s) = bs + b,s and s , , s 2 a re the r o o t s of g(s) = 0 .
-Xx (B) By using the transformation Y= e y, X Φ 0, and assuming that
(3.15), (3.16) and (3.17) hold, then the operator equation:
I~a(a + bx)l~k eVX J"1 (a + bx)k e~VX Ja_ 1 eXx y = 0 (3.18) xQ xQ xQ
i s e q u i v a l e n t to the second o rde r l i n e a r d i f f e r e n t i a l e q u a t i o n :
( a + bx)y" 4- \_al-2aX) + [b χ - 2bX) x]y' +
[aX2 - a λ + dx + (Ζ?λ2 - b \)x]y = 0 . (3 .19)
(C) The differential equation
(a + bx)y" + (^ + B x)y' + AQ + S Q x ) y = 0 , (3.20)
Z?^0, |5χ | + |5Q | >0, is equivalent to (3.18) if a
G'aj
μ = i- (3.21) G;(X^.)
bF'X.) -aGfX.) F(X.)
b2 G'X.)
i = 1,2), λ , λ are the roots of the quadractic equation G(X) = 0, G(X) = bX2 +B1X + BQ and F(X) = aX2 + ΑχΧ + AQ.
Details of these properties are found in [8].
Some Applications of Generalized Calculus 69
2 . SOME METHODS OF SOLUTIONS AND PARTICULAR EQUATIONS OF SPECIAL FUNCTIONS
2.1 Some Methods. The representation of differential or integro-differen-
tial equations by operator equations may simplify the problems of obtaining
solutions of such equations. This is made possible by the use of the opera
tional properties of the integro-differential operator of generalized order
[1]. In applying some of these properties to the operator equations, one may
find their solutions. These solutions are satisfied by the equivalent dif
ferential or integro-differential equation. fXu \ In the process of solving an operator equation a term of the form ί .0! \a J
appears in the solution. When a-n is a positive integer this term has been
defined as follows: If F£C on [a,b] and x_ I n F = DHF = 0 a
then x JA C Λχ-α) 1 - ° = F = Σ , x · a u T(n-t+l)
where C. are arbitrary constants. This definition has been extended ([1], p. 20) to include all values of a where Reot+n>0. In this case the definition takes the form
n- 1 α . η = τ?= V r (χ~α) I -0 = F = ) C. X^J±I . (4.1)
a ΐ-ο Γ(-ο-*)
Equation (3.1). If we consider the operator equation (3.2) representing the second order linear differential equation (3.1) and they are equivalent to each other we would find, by operating on and multiplying both sides of (3.2) by the operators and the quantities:
x IW , a + bx)P~l, (a + dx)q~l . . . . e t c . ,
that χ X0
Yte) = I~W+1a + bx)~pc + dx)~p (a + dx)~q
\
'K + / (c + dxf~l (a + bx)P~l (lW-o) , ( 4 ' 2 )
where K is an arbitrary constant, w is not a positive integer, and I ·0 may be determined as indicated by (4.1) after defining the lower limit of the interval of integration which is usually chosen from among the singular points of the equivalent differential equation (see [2], p.52). It is clear that (4.2) is a linear combination of the two particular solutions
70
ΥΛχ) = Κ I1 W(a + bx) or
M. A. Al-Bassam
"P (c + dx)'q
YAx) = I1 W(a + bx) ~P (o + dx) q x I (a+bx xn
\P-i (o + dx q-' (iw-o) V ;
each of which may be easily shown to satisfy (3.2) and consequently they satisfy (3.1).
Equation (3.20). The representation of this equation by the operator equation (3.18) may be used to solve the equation by the operational method. By operating on and multiplying both sides of (3.18) by:
respectively we have
[a + bx k-\ -\\x
, X
y (x -X) = e I (a+bx)
X
Ί -k ]ix e
\K+ [ e-^(a + bt)k-l(l L J \X,
0 J dt (4.3)
0 where K is an arbitrary constant and ί I ·0ί may be determined after defin-
0 ^ ing the lower limit of integration ([.2], p.52, [3], [8]). It may be noted that (4.3) is a linear combination of the two particular solutions
-k \\x e x
y (x ; λ ) = Ke I (a + bx) 1 x
0 / Λ Ν λχ 'ι-οΐ/ . , x -k \ix, y2(x;X) = e I (a + bx) e^ (
xQ
+ ba "' (£·») ·
(4.4)
(4.5)
These solutions satisfy (3.18) and consequently (4.3) satisfies (3.20). Remark, In most of these equations solutions y may be considered as the generalized Rodrigues formulae. 2.2 Particular Cases
(1) Confluent hypergeometric equation. The hypergeometric differential equation together with the more general forms of differential equations such as Riemann — Papperitz and Gauss's types have been discussed in a detailed study in a previous paper [2]. So, it is appropriate to present, as an example, this particular equation. This equation is given by
xy" + (ß-ar) y' - yy = 0 . (4.6)
G(X)
By comparing this equation with (3.20) we notice that F(X) = βλ— γ = λ2 — λ= 0 and the roots λ ^ Ο and X = \ .
Some Applications of Generalized Calculus 71
(A) The case when λ = 0 . We have μ= 1 , α = γ and k= B ~ Y . Thus the
e q u i v a l e n t o p e r a t o r e q u a t i o n may be w r i t t e n as
( A . l ) j - Y χ γ " β + 1 ex r 1 x*^ e~x ly~l y = 0 , xo xo xo
whose s o l u t i o n may be g iven by
(A. 2) y(x;0) = y^x-,0) -l· y2(x;0) ,
where x
y(x;0)=K^ J 1 _ Y eX χΊ~^ = £ χ 1 " 3 F. ( γ - 3 + 1 ; 2 - 3 ; a;) I 1 Q 1 1 1
z/ o ( x ; 0 ) = J e x ' M I e x [ I · 0 ) = K F ( γ ; 3 ; x) , 2 0 0 V 0 / Z l !
where X is an arbitrary constant, the lower limit is determined to be zero and
2 Γ(γ) o with X2 being an arbitrary constant.
(B) The case when λ = 1 . In this case we have μ = - 1 , α = 3 _ Υ , k = γ and the equivalent operator equation may be written as
/D n τΎ"3 ι-γ -χ ^-ι γ χ ^3-γ-ι -χ (B.1) Γ' χ e I χ e I e y = 0
o o o which has a solution of the form (B.2) y(x; 1) = y (x ; 1 ) + y ^ (x ; 1 ) W n e re X _ Q + l _ l_ß w, (x; 1) = C, ex IY ^ x T g " x = C x P 1F1 (γ-3+l ; 2-3;x) 1 L o i i
, .v χ ^ γ - 3 + ι - γ -x r x γ - ι ( ΐ 3 ~ Ύ · 0 ) y (x\\) = e I x e l e x \Q ) 0 °
= D (γ;^;*) where C is an arbitrary constant, β,γ are assumed to be parameters with ranges determined by the validity of the operations and the properties of function involved,
l ß - Y . 0 = ( ß - . ) D ^ Ü , and c . c H ^ X i o Γ(β-γ) i Γ(2-β) D is an arbitrary constant.
Details of this work may be found in [8] .
72 M. A. Al-Bassam (2) Legendre Differential Equation. The equation
(1 - x2) y " - 2xy' + a(a + \)y = 0 (4.7)
is a particular case of (3.1) where a = o = d=\,b = -\9a=09 bY= -2 and d =α(α+ 1). The roots of the quadratic equation
w2 - w - a(a + 1) = 0
a re w = a+ 1, w2 = - a , and p = q - - a , p = q = a + 1 . Therefore the two
o p e r a t o r e q u a t i o n s
- a - 1 , 2 . i \OL+l ? - l (* 2 - i r I xn
x z2-\ ) a J a y = 0 , (Re a > - I]
* 0
(4 .7 ) ,
J a ( x 2 - i ) " a / " M * 2 " ! ) ' 0 , (Re a < 0) (4 .7 ) .
are equivalent to (4.7) whose solutions are solutions of (4.7) or (4.7)?. If (4.7) is taken then its solutions may be given by
x -α , 2 y(x) = K. I " xz-\ 1 -L ^
(4.8)
x x yAx) = XI u (x2-!)" I (x2-l)"
" #„ #'„ (4.9)
where X and X are arbitrary constants. It has been shown ([3], p.96), that y (x) and y (x) are the Legendre functions P (x) and Q (x) respectively. Equation (4.8) may represent the generalized Rodrigues formula for the solution of Legendre equation which can be written as:
yi = „a 1 r " a / 2 i \c
I [χΔ-1 )
We find that
2>i =Paix) = Γ(2α+1)χ
2α[Γ(α+1)]'
2"Τ(α+1) ±c
Ja, - Ja + J ;
i - a ;
(4.8),
(4.8),
which for a = n (positive integer) is reduced to P (x) . Also we have
y2 = Qax): /ττ Γ(α+1 ) 0α+ι α+ι 7. . : 2 χ > (α+; 2F1
J(a+D, J(«+2) α+| *2
For further details see [5].
(3) Bessel's differential equation. By using the transformation y=x z the equation
x2y " + xyf + x2 - v2) y = 0 (5)
S o m e A p p l i c a t i o n s of G e n e r a l i z e d Ca lcu lus 73
can be written in the form xz" + ( 1 + 2\>)z' + xz = 0. (5.1)
Comparing this equation with (3.20) we find that a- 0 9 b- \ 9 A =2v+l, B1=0, AQ = 0 and BQ = 1 . Also we observe that G(X) = λ2 + 1 , F(X)=(2v+l)X and the roots of £(X) = 0 are λ, = i and λ2 = -i, (i2 = - 1 ) .
(A) The case when \ = i . In this case, by (3.21) we find that a = v^i, μ = -2i and k=V+-. Thus the equivalent operator equation may be written as ,A 1Λ
XT-(v + i) J-v -2£c *-l v + J 2tx *v-i -ix (A.l) J ^ x e I x d e I e 2 = 0 , 0 0 0
which has the solutions of the form
(A.2) z (x; i) = z (x ; i) + z (x ; i) ,
where , .s v ίχχΛ-ν -2ix - ( v + J ) z x; τ) = K e I* e x
1 i 0 t . , ix *\-v -2ix - ( v + J ) f v - J 2 ^ ( l V + J . 0 j
0 0
(X i s an a r b i t r a r y c o n s t a n t ) .
I f v = n + J , (n i s a p o s i t i v e i n t e g e r ) , we have
(A.3) zx(x;i) = X 1 ( - 2 ) n x " n " ^ 0 \-iJn + f(x) + ( " D * ^ ^ (a)
/ ·\ M o"n"l „i -n-\ \J _,_, (a:) — (-1) tJ" x (x) ,
from which we obtain the solution of (5) for this case which may be given by
(A.4) y(x;i) = Ll J ^ (x) + Μχ J_n_^(x) ,
where L and M are arbitrary constants.
(B) When λ = - i , we find that a = V+J , k = X>+\ , μ = -2i and the equivalent operator equation takes the form
/τ> 1λ *-(v+*) J-v 2 ^ *-i v + * -2ia: ifv-J ix n (B.l) I 2' x2 e I x * e I 2e 2 = 0 , 0 0 0
whose solutions may be given by
(B.2) z(x;-i) = z (x;-i) + z (x;-i)
where ix . f i - v lix - ( v + J ) I *- e x z(x;-i) = C e X I 2 v e"
1 1 r 0 r / v+1 \ . , ix X
T\-v 2ix - ( v + J ) ^ - 2 7 ^ V - J T V ? - 0 z ( a ; ; - t ) = e X2 e x 2 i s a: ^ \ 0 /
2 0 0
74 M. A. Al-Bassam where C is an arbitrary constant. Also when v = n+|, then we find that the solution of (5) for this case is given by
ij(x;-t) = L2 Jn+i(x) + M2 J_n_iM
where L and M are arbitrary constants. For details see [8] .
(4) Equations of Laguerre type. In this case certain types of Laguerre differential equations are discussed and solutions are obtained in the form of Laguerre function of generalized type [7] which may be called as Rodrigues generalized formulae for Laguerre functions. These solutions are obtained by dealing with the equivalent operator equations as indicated above.
Consider the second order differential equation
xu " + [ ( 2 λ + μ ) x + 3 + 1 ] u1
+ [λ (λ + μ) x + 3(λ +μ) +μ(α+ 1) + λ] ?Λ = 0 , (6.1)
where λ, μ, α and 3 are parameters. This equation can be easily shown to be equivalent to the operator
equation - a - i a + 3 + i - y x ' - i - a - 3 yx ^a 3 λχ _ fc ΟΛ
I x e l x, e I x e u = 0 9 (6 .2) 0 0
whose s o l u t i o n may be given by
0
where
u(x) = u (x) + u2(x) ,
X f N 7/ _ 3 - λ χ - α - y x α+3 ,, ΟΛ u Ax) = K x e I e x (6 .2)
i 1 0 1
f x 7/ ~3 -λχ - α -\ιχ α+3 ^ - α - 3 - ΐ μχ , ( 6 . 2 ) . \ι (χ) = Κ χ e I e x Ix e 2
2 λ 0 0 K and K a r e a r b i t r a r y c o n s t a n t s . 1 2 y
Equation (6.2) represents the generalized Rodrigues formula for the (3) solution of (6.1) which may be denoted by L ( A ; y ; x ) , for if we put
K = 1/Γ(α+1) we would get CO ]ζ
( Λ Γ(α+β+1) -(X+y)x V ("α)k ( μ χ ) , V X ; - Γ(α+1) Γ(β+1) " L ( 3 + i ) r (k+l)
k = o k
where (a) = α (α+ 1 ) ... (a + r- 1 ) , (α) = 1 . Thus we find that
Γ ( 3 ) / Λ χ Γ(α+3+1) -(y+X)x „ . 0 . , L (λ;μ;χ)= e Μ . F. (-α ; 3+ 1 ; yx) . α Γ(α+1) Γ(3+1) ! 1
Some Applications of Generalized Calculus 75
P a r t i c u l a r cases of (6 .1)
(A) Let ß = 0 in ( 6 . 1 ) . Then we have the d i f f e r e n t i a l e q u a t i o n
(A. l ) xu" + [ ( 2 λ + μ) x + \u' + [ λ ( λ + μ ) χ + μ ( α + 1 ) + \]u = 0
with the equivalent operator equation
,A ΟΛ -α-ι a+l -\ix -l -a \ix ^α χ _ (A.2) I x e l x e I e u = 0 .
0 0 0 We notice that the solutions (6.2) and (6.2) represent solutions of
(A.l) when 3 = 0. Furthermore we observe that the R.odrigues formula for the solution of (A.l) is L (X;u:x).
(B) If λ= -1, μ = 1, then we would have Laguerre differential equation
(B. 1) xu" + (1 + 3-x) u1 + au = 0
with its equivalent operator equation
,., ολ ^-α-ι α+3+ι -x -\ -α-3 χ τα 3 -x n (B.2) I x e I x e I x e u = 0 .
0 0 0 Equation (B.l) is satisfied by the solutions (6.2) and (6.2) with λ = -1 and μ= 1. In particular, the Rodrigues formula for the solution of (B.l) has the form:
r(3)/ ! , x 1 -3 x r-a -x a+3 L (-\;\;x)= x e I e x a Γ(α+1) o
It may be pointed out that the Rodrigues formula presented here is in its generalized form, for if a = n, a positive integer, we have L (-1 ; 1 ;x)
which is of the same form as mentioned in references of Special Functions. Simple Laguerre differential equations and their equivalent operator
equations together with their solutions follow in the same way from the above. Remarks. (1) In this article, many cases of integro-differential equations have not been discussed. For further study it may be of interest to see [10] in the bibliography.
(2) The method applied here may be applied to all types of equations whose solutions are some types of special functions. In fact the writer is studying some operator equations which are equivalent to a generalized form of integro-differential equation of which most of the equations of special functions are particular cases.
76 M. A. Al-Bassam REFERENCES
1. M.A. Al-Bassam, Some properties of Holmgren-Riesz transform, Ann. Scoula Norm. Sup. Pisa 15 (1961), 1-24.
2. M.A. Al-Bassam, Concerning Holmgren-Riesz transform equations of Gauss--Riemann type, Rend. Circ. Mat. Palermo XI (1962), 1-20.
3. M.A. Al-Bassam, On certain types of H-R transform equations and their equivalent differential equations, J. Reine Angew. Math. 216 (1964), 91-100.
4. M.A. Al-Bassam, Some existence theorems of differential equations of generalized order, J. Reine Angew. Math. 218 (1965), 70-78.
5. M.A. Al-Bassam, On an integro-differential equation of Legendre-Volterra type, Portugal. Math. 25 (1966), 53-61.
6. M.A. Al-Bassam, Existence of series solution of a type of differential equation of generalized order, Bull. College Soi. Univ. Baghdad 9 (1966), 175-180.
7. M.A. Al-Bassam, H-R transform equations of Laguerre type, Bull. College Sei. Univ. Baghdad 9 (1966), 181-184.
8. M.A. Al-Bassam, On Laplace's second order linear differential equations and their equivalent H-R transform equations, J. Reine Angew. Math. 225 (1967), 76-84.
9. M.A. Al-Bassam, On the existence of series solution of differential equation of generalized order, Portugal Math. 29 (1970), 5-11.
10. M.A. Al-Bassam, On some differential and integro-differential equations associated with Jacobi's differential equation, ./. Reine Angew, Math, 288 (1976), 211-217.
11. M.A. Al-Bassam, Some operational properties of the H-R operator, Ann. Scoula Norm. Sup. Pisa 27 (1973), 491-506.
12. M.A. Al-Bassam, H-R transform in two dimensions and some of its applications, Fractional Calculus and its Applications, Proceedings of the International Conference, Univ. of New Haven, June 1974, edited by B. Ross, Lecture Notes in Mathematics, (457), Springer-Verlag, New York, 1975.
13. M.A. Al-Bassam, On fractional calculus and its applications to the theory of ordinary differential equations of generalized order, Lecture Notes in Pure and Applied Mathematics, Nonlinear Analysis and Applications, Vol. 80, Marcel Dekker Inc., New York, 1982.
14. H.T. Davis, The Theory of Linear Operators, Principia Press, Bloomington, Indianna, 1936.
15. A. Mambriani, The derivative of any order and the solution of hyper-geometric equation, (translated title), Boll. On. Mat. Ital. (2), (1940), 9-10.
16. A. Mambriani, On the partial derivative of any order and the solution of Euler and Poisson equation, (translated title) , Ann. Scoula Norm. Sup. Pisa 11 (1942), 79-97.
17. K. Nishimoto, On the fractional calculus, Dissertations in celebrations of the 30th anniversary of College of Engineering, Nihon University, Japan, (1977).
18. K. Nishimoto, On the fractional order's differential equation wv + wvl>g = f, J. College Engrg. Nihon Univ. 25 (1984), 47-52.
19. K. Nishimoto, S. Owa and H.M. Srivastava, Solutions to a new class of fractional differintegral equations, J. College Engrn. Nihon Univ. 25 (1984), 75-78.
20» K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
A BOUNDARY VALUE PROBLEM FOR HEAT AND MASS TRANSFER IN POROUS MEDIA
Y. ATIK ENS, V. Kouba, Algiers, Algeria
Introduction. A model to study the nonisothermal time-dependent flow of two immiscible fluids in nonhomogeneous anistropic porous media is presented. This model is inspired from that given in [1] in studying the isothermal flow of two immiscible fluids in porous media. Our motivation is to study oil recovery by hot water injection. We are interested in temperatures less than 300 C and pressures in the range 100 — 300 at. Under these conditions the water remains liquid [9]. In [10] one can find a modeling for oil recovery by steam injection. In this paper we study a simplified version of the model namely the nonlinear system
Γ lx)^j+ 7-V0 =div(AV6)
\ mx)~ = - divVl , -νλ = KQa VS - bV + P
I d i v V= 0 , -V = XVP + f
in a r eg ion Ω = Ω χ ] 0 , T[cz IR χ IR, where Ω i s a bounded domain; 1 (x), m(x),
a(x,Q,S)>0, b(S) , F(x,Q ,S) , and f O , 9 , S ) a r e given f u n c t i o n s ; Λ ( Θ , £ ) ,
K (x), and K(x, Q ,S) a r e p o s i t i v e d e f i n i t e t e n s o r s . The unknown func t i ons
0 ( x , t ) , S(x,t)9 and P(x,t) a r e r e s p e c t i v e l y the t e m p e r a t u r e , the reduced
s a t u r a t i o n , which must s a t i s f y 0 < S < 1 , and the reduced p r e s s u r e . The func
t i o n a v a n i s h e s for 5 = 0 and 5 = 1 , t h i s means t h a t the second equa t ion i s
d e g e n e r a t e . We cons ide r t h e boundary and i n i t i a l c o n d i t i o n s ( 9Ω = Γ0 U Tj U Γ 2 ) :
7(Θ,5 ,Ρ) · n = V(Q,S,P) -n = AV6-n = 0 , (x , t ) € YQT = YQ X [0,T]
V'n = Q(x,t), νλ ·η = bQ(x,t) , U , t ) £ Γ 1 ? Ι , Q given
S = SQ(x,t) , P = P 0 C r , t ) , ( x , t ) e r 2 y and θ = θ0 ( * , £ ) , (x , t) € Γ ^ U Y 2T
11
78 Y. Atik S = SQ(x) , θ = θ0(χ) for t = 0 and χ € Ω . Under suitable assumptions on the given functions and tensors (see
sect. 3) we prove the existence of a generalized (bounded) solution of this system with the above boundary and initial conditions. For this purpose we will use mainly the method (of compactness) given in [1]. As this reference is not easily available we reproduce here some of the calculations given in it.
1. The System of Equations Governing Heat and Mass Transfer in Porous Media. 1.1 The flow of two immiscible fluids (e.g., oil and water) in a porous medium is characterized by the velocities V^ and V2 , the pressures P, and P„, the densities p1 and p2 , and the saturations 5 and S of the two liquids. As we are interested in the nonisothermal flow, we take in addition the temperature Θ of the two liquids. In each point of the porous medium we assume that the two liquids and the solid porous matrix are at the same temperature. We will reserve the subscript 1 to the wetting liquid and the subscript 2 to the nonwetting one (e.g., 1 for water and 2 for oil).
The above flow's characteristics are related by the following well-known relationships (see, e.g., [2]): (a) The equation of continuity (mass conservation)
^ ( m p i S i ) + d i v ( p i ^ ) = 0 , i - 1 , 2 . (1.1)
where m = mx) is the porosity if the medium. (b) Darcy's law
K V. =~Kn — ( V P . - Q.g) 3 i = l , 2 . (1.2)
where K = K (x) is the filtration symmetric tensor, K~. is the permeability of the medium to the liquid t, it depends on saturations, y. is the dynamic viscosity of the liquid i, > it depends on temperature, and g is the gravitational acceleration vector. (c) The capillary pressure, defined by
o _p 2 1
where (1.3)
1 1 e [ o , i ] i -s?-s°
is the "reduced saturation" of the wetting liquid (e.g., water), and S. , i = 1,2, are the residual saturations.
The capillary pressure is given by
/ - ^ / 2
Pc(x,S) = Pa(x)j(S) , Pe(x)= σ i-^-j cosY, (1.4)
Heat and Mass Transfer in Porous Media 79 where σ is the interfacial tension, which is constant for any pair of substances, γ is the contact angle, \K | is the absolute value of the determinant of K , and J(S) is the Leverett function.
The Leverett function and the relative permeabilities K .(S) are given experimentally; they are subject to the hysteresis phenomenon.
The liquid i can flow until its saturation S. reaches the residual saturation S. , then, it ceases completely to flow so that
Kois\) = 0 , i= 1 ,2. (1.5)
(d) The two liquids fill the entire pore space:
SY 4- S2 = 1 . (1.6)
(e) The equation of conservation of thermal energy. The equation of conservation of thermal energy for all components may be written as (no source) [ 7 ] !
ltt\piei) + d i v ( I p i e i ^ ) = - d i v ^ ( , · 7 )
-z> l i=\ where T=mS , τ 0 = mS , and τ = 1 —rn. Here e . is the internal specific 1 1 ' 2 2 3 ^
energy of the component i9 ^ = 1 , 2 , 3 ; the subscript 3 stands for the porous matrix, and q is the heat "flux" vector which is given by (f) Fourier's law
q = -AV0 , (1.8) where Λ = Λ(θ , S) is the thermal conductivity tensor of the porous matrix--liquids system. (g) To the previous equations, one must add the equations of state:
p. = p. (P. ,θ) , v i = 1 ,2. We assume in this work, that p . = c o n s t . , i - 1 , 2 .
The coefficients of the above equations, i.e., the functions and tensors in , K , Kn . 9 u . , i = 1 , 2 , Λ , and J are considered to be known. m and K depend only on x , K . and J depend on S, μ. , i = 1 , 2 , depend only 0 (U ^
on Θ, and Λ depends on Θ and S. The other coefficients p., S . , σ and γ are constant.
We choose the dependency of μ and μ on Θ such that μ,(θ)
^ = constant 100 < Θ < 300 . (1.9) μ (Θ) 2 We put K s)
0^ v It follows from (1.5) and properties of KQ . and μ^. that
80 Y. Atik
k . ( Θ , 5 ) > Ο , vs e ] o , i ] ve e IR , ] 0 t > ( i . i o )
^ ο ι ( θ ' ° ) = ^ 0 2 ( θ ' 1 ) = ° > ve e ]R . J We consider the case where S is constantly increasing (the wetting
fluid displaces the nonwetting one) therefore, as experimentally known [2]: The function k^ is increasing, kQ2 is decreasing, and P decreasing with respect to S (then —-r- < 0j .
1.2 Boundary and Initial Conditions. We assume that the flow occurs during the time interval 0 < t < T in a three-dimensional domain Ω with smooth boundary Γ, where Γ = rQ U Γχ U Γ2; Γ0 , Γχ , Γ2 quasidisjoint. We put YiT = Γ\ X [Ο,Τ] , i = 0, 1 ,2, and Ω = ΩΧ ]0,Τ[. We denote by n the outward (from Ω) unit normal to Γ , x = (x, , x2, x3) the space variable, and t the time variable. (a) Boundary conditions:
* On Γ the discharge of the mixture is known as
($!+%) -n = Q(x,t) , x9t)eTlT . (1.12) * On T2 the pressure and the saturation of the wetting liquid are known
a S P ^ P ^ i x t t ) , S=SQ(x9t)9 (x9t)€T2T. (1.13) * Γ is impervious to the two liquids:
ΪΛ -n = 0 , (x,t)er0T, i= 1 ,2. (1.14) * also, no heat flux through Γ :
AV0-n=O, (x9t)EYQT. (1.15)
* On Γ1 UT2 the temperature is known as
θ = QQx9t) , x9t)erlTi>T2T . ( i . i 6 )
(b) Initial conditions'. S(x90)=Sx), Q(x90) = Q(x) , x t t t , (1.17)
REMARK. Neglecting the normal component to Γ of the gradient of the capillary pressure, we have on Γ : \7Ρχ ·Ή = VP ·η, and using Darcy's law and neglecting the gravity, we get on Γ
Hence, (using (1.14))
V--n = y-^t— Q(x9t) , i = l , 2 , x,t)eV (1.18) "01 02
Heat and Mass Transfer in Porous Media 81
2. Transformat ion of E q u a t i o n s .
2.1 Let V be 7= 7χ + 72 ( f i l t r a t i o n v e l o c i t y of the m i x t u r e ; \V\ g ives
the s p e c i f i c d i s c h a r g e of t he m i x t u r e ) . As p 1 and p 2 a r e assumed c o n s t a n t
and m = m(x) 9 u s ing (1 .1) and ( 1 . 6 ) , we o b t a i n
d iv 1/ = 0 . (2 .1)
Let us i n t r o d u c e the "reduced p r e s s u r e " [1] (of the w e t t i n g l i q u i d ) by
P(x9t) = Pxx9t) - I 2)0(ξ) -φχ,ζ) άζ + PlGx3 ,
S(x,t) where b =k Ik wi th k=kQl + kQ2, and gVx = - g . According t o (1 .9 ) b
does not depend on Θ, i t depends on S. Taking i n t o account (1 .2 ) and (1 .3 )
we can w r i t e V in terms of 6 , S , and the g r a d i e n t P . In f a c t
_ 7 = # V P + f (2 .2 )
where k = kK , and PIP
? = ? ( a : , e , 5 ) = x J f c 0 v ( ^ ) d e + fc02^0VicPö+fc02(pi-P2)Ä0^
Here the symbol V means t h a t in c a l c u l a t i n g t he g r a d i e n t of P we ignore
the p resence of x in S .
I t i s a l s o easy t o see t h a t
-Vl = KQaVS -bV +F , (2 .3 )
where
b=kQi/k, F = F(x,^S)=-kQiko2k-lKo[VxPc+(pi-p2) g],
3P a = a(x,Q,S) =-k01k02k~l -— = aQ(S) Pcx)/v2(Q) .
do Here „ „
„ (a) 0 1 °2 dJ(g) Vs' - iTx ds ·
According to (1 .10) and (1 .11) we have
ax,Q,S) > 0, VS € ] 0 , 1 [ , ax ,Q ,0) = ax ,Q,\) = 0.
F i n a l l y , u s ing ( 1 . 1 ) , ( 2 . 1 ) , (2 .2 ) and (2 .3 ) we o b t a i n t he system
f m 4 T = d i v ( Z a V 5 - &F+ F) (2 .4 )
d iv(XVP + / ) = 0 , -V=KVP + f , (2 .5 )
where m =m(\ - S^-S^)
2.2 Let us transform equation (1.7). We have
•£=1 ^=l
82 but
3 τ ι . + T F + d l v 7 i
Y. Atik
3τ0
^t- + d i v F 2 0
a l s o , as the porous m a t r i x i s immobile, and m = m(x), we have
3 τ - r f + d i v 7 3 = £ (l-iä) + div73
On the o t h e r hand de.
%_ 8Θ
^ at Ve. ι V0 ,
0 .
i= 1,2,3.
where c.(6) = (3β·)/3θ is the specific heat capacity of the component i at "Z- 'Z'
temperature Θ. Then, using (1.8) we have (2.6)
where I-
Z H + K pi V °2 Ρ2^2]"νθ =div (Λνθ) '
We observe that Z depends on χ9 Θ, and S.
We are interested in the temperature range [100 , 300] where c and c_ are nearly constant [6]. Then we may assume that t does not depend on Θ. We also assume that 1 does not depend on S: I = I (x) . Finally we assume that we can write
Px alVl + P 2 ^ 2 7 2 ^ e 7 , (2.7)
where e is a constant. To simplify notation and without loss of generality we may assume that e = 1 .
In a subsequent work we let Z- to depend on S and will not make use of (2.7).
2.3 The system to solve, therefore, will be
l(x) | | + 7 · ν θ = d i v ( Λ ν θ )
W(a;) | £ . = d iv (X 0 aV5 - £ 7 + F
[ d iv (XVP + f ) = 0 , - 7 = KS/P + f
with the boundary and i n i t i a l c o n d i t i o n s (1 .12) —(1.17) where (1 . 12) — (1 . 14)
a r e modif ied as fo l lows
* 7 ( Θ , 3 , Ρ ) · η = Vx(Θ,£,Ρ)·η= 0 ,
* 7 ( θ , 5 , Ρ ) · η = £ ( * , £ ) , 7 1 ( e , S , P ) - n = fee(a:,t) , ( a ; , t ) e r i r ( 1 . 1 3 ) '
* 5 = 5 0 ( x , t ) , Ρ = Ρ ( Ρ χ , S ) = Ρ 0 ( * , £ ) , ( x , t ) e r 2 y . ( l . l 4 ) /
From now on, we refer to the system 2.3, its boundary and initial conditions by symbol I .
(x9t)evQT (1.12)'
Heat and Mass Transfer in Porous Media 3 . Genera l ized S o l u t i o n s of System I.
3.1 Assumptions on the Data of System J . We assume t h a t t he da t a m(x) ,
l(x)9 KQ(x), Pc(o:), kQ^(Q,S)9 and Λ ( θ , 5 ) a r e def ined for
x9Q9S) Ε Ω Α = Ω Χ 1 + Χ [ 0 , 1 ]
and s a t i s f y the c o n d i t i o n s :
83
( i ) * K and Λ a re symmetric ,
M~l < (1 ; m ; k ; dP
ZS ; ( Χ 0 ξ , ξ ) / | ξ | 2 ; ( Λ ξ , ξ ) / | ξ | 2 ) <Μ ,
uniformly in ΩΑ , where M i s a p o s i t i v e c o n s t a n t ,
* k01(Q,S), k 0 2 ( e , S ) . , ^ f - , g , and Λ ( θ , δ ) a r e con t inuous wi th r e s p e c t t o Θ and S, and
Sup kQl ; K Q 2 |
dJ\\ \dS\
~,1R + X [0,1] ( θ , 5 ) €1R + X [ 0 , 1 ]
< M , II P
& 0 1 ( θ , 5 ) ; Α : 0 2 ( θ , 5 ) < M ,
n | | < M , (μ ; μ 9 ) > μη > 0 for θ > 0 . C[0, 1] " 0 , Ι οο , Ω
l 2 °
( i i ) 0 < ( a ; / c 0 1 ; f e ) for ( Θ ,S) € B + X ] 0 , 1 [ and
α ( χ , θ , Ο ) = α ( χ , θ , ΐ ) = ^ 0 1 ( θ , Ο ) = ^ 0 2 ( θ , ΐ ) = 0 .
The boundary d a t a Θ ( x , t ) , S (x , t) , and P Q ( x , t ) a r e assumed to be
def ined not only on t h e boundary but a l s o on Ω = Ω X [ 0 , T) and s a t i s
fy , wi th Q(x,t) for (x,t)ET9φ , t h e c o n d i t i o n s : IT
( i i i )
f \ \ d Q o dso
(x90)=QJx) , SJx 90)=SJx) for xEQ, and
2 , - , Ω σ V I I 1 F - T F I I , o M l v e o ; V 5 o l ->iP»>™« 2,ςισ "2,Ω^
Ιοο,Γ IT U,~ ,r 1 T ; < M, where q > 2 .
Here · deno tes the LAü ) norm and II II 2 o 2 T Ι^,οο,Ω^ the L0 (Ω-.) norm.
( i v ) 0<QQx9t) < Μλ a . e . in Ω , M 61R+ > and
0 < 6Q < 5 Q ( x , t ) < 1 - 6 a . e . in Ω , 6Q and 6 χ e ]R+
3.2 REMARK. The conditions (i) imply
*n (a/(feQifco2)) , ?/fcQ2 , ^ / ( ^ 0 1 ^02) j < M 0( M ) ' (*> Θ><5)£Ω Α, and
(|| K || , || TV ]| ) < Af . (||*|| denotes matrix norm.)
3.3 DEFINITION. A triplet ( Θ , 5 , Ρ ) , of measurable and bounded functions in Ω φ , is called a generalized solution of system I if :
84
(a)
(b)
(c)
(d)
Y. At ik
0 < 6 ( x , t ) and 0 < Sx,t) < \ a . e . in Ω .
ν θ € £ 2 ( Ω Γ ) , aVS£L2(ttT), and VP€ L2 >co (Ω^) .
6 = e 0 ( x , t ) fo r ( x , t ) e r i y U r 2 y and S = S0(a:, £ ) , P = P (a?,£)
for ( j ? , t ) e r IT
, 1 » 1 ν η € ί / 2 · (Ω^ , r i ? 7 U r 2 T ) n ^ p , T)(x,T)=0, x£Q
A »1 ν φ € 7 2 ' (ςΐτ,Τ2Τ),φ(χ,Τ)=0,χ£ΪΙ
ν ψ ^ ( Ω , Γ 2 ) we have
d . l ) - ( Ζ θ , η ) π + ( 7 , η ν θ ) π + ( A V 0 , V n ) 'έ'Ω, Ω„ Ώ„ ( * θ 0 , η ) ,
' τ 06τ "τ uuo d - 2 ) ^ 5 ' ^ ) Ω τ
+ ( ^ ΐ ' ν φ ) Ω τ = ( ^ ' φ ) Γ ι Τ - ( ^ ο ^ ) Ω ο
d .3 ) ( 7 , ν ψ ) = ( β , Ψ ) Γ a . e . in ] 0 , Τ [ ,
where η = (3η ) / d t , ( · , ·)<^ deno tes the s c a l a r product of L (Ω™) ,
( · , · ) Γ the s c a l a r product of L (Γ ) , and Ω = Ω χ θ . Here
du du du
and
wl»l / ^ r> \ J du du du du c _ / r . x ι n \
and '
« $ ( a . r 2 > - « : « . ^ , ^ . ^ Μ » ) ; «|Γ - ο REMARK. Since ^ = - (£0 aVS- &7 + F) , and
(2>v,v<p; 0 - ^ , , ^ , τ . -(i-V'VS.ip)
here 2?« = (db)/dS, (d.2) may be written as Titten £
d.2)' - ( ^ 5 , ψ ^ ) Ω + (£0aVS + F,Vcp)fi + (fcc 7- S/S ,<p) 0 = ( ^ 5 η > ψ ) (
Our main result is the following: S Ώ„
THEOREM. If condition (i)—(iv) are fulfilled system I has at least one generalized solution, i.e. there exists at least one triplet (Θ ,S, P) satisfying (a), (b), (c), (d.l), (d.2), anc[ (d.3) of definition 3.3.
To prove this theorem we will construct a triplet with the required properties; this needs several steps:
3.4 Regularization. We extend the coefficients of identities (d.l), (d.2)', and (d.3) by putting 'fx,Q,S)
/(*,θ,0) f(x,0,0)
f(x,09\ ) ./(ί,θ,Ι)
(ar,0,S) € Ω* Θ > 0 , S < 0 Θ < 0 , S < 0 θ < o , se[o,i] θ < o , s > l Θ > 0 , S > 1
Heat and Mass Transfer in Porous Media 85 ->-
a l s o : α ( χ , θ ,S) = α Λ ( χ , θ , 5 ) + ε , ε > 0 ; 7 , = — 7 _ , /z > 0 . we have ' '
\Vh\ < \V\ and \vh\ < \/h . (3.1)
4. Generalized Solutions of the Regularized System I , . 4.1 DEFINITION. A triplet (Θ,£,Ρ), of measurable and bounded functions in Ω , is called a generalized solution of the regularized system I , if :
(α) Θ,5,Ρ arein ^(Ω^) and V P E L (Ω^). (3) Θ,£,Ρ satisfy conditions (c) of defintion 3.3. (γ) For all η, φ, and ψ chosen as in (d) we have Y.l) - ( Z 0 , n t ) ^ + ( ^ , n V e ) % + ( A . V e , V n ) ^ = ( ^ o , n ) ^ (4.1)
γ.2) ~mS^t)ü +(Ζ0αν,5 + ? Α , ν φ ) Ω + (d ϊ^ · VS ,ψ) Ω = (mSQ^)q (4.2)
γ.3) (7,νψ)Ω= (ρ,φ)Γ a.e. ίη]0,Γ[, (4.3)
where d=(b ς)Λ» a^d
vrai max 0 < t < T
M < - » * > l l 2 , « + I H 2 , n r< + ~ -
We are going to investigate first the regularized problem, then, passing to the limit with respect to h and ε , the proof of the existence of one generalized solution of the degenerate system I is achieved.
4 . 2 THEOREM. Suppose conditions (i-iv) are fulfilled and let ( Θ , S , P ) be a generalized solution of I , . Then
0 < a = min Q (x,t) < Q(x , t) < a, = max Θ (x , t) a.e. in_ Ω , (4.4) A lm Λ im
0<S(x,t)<\ a . e . in Ω , . ( 4 .5 )
I f , in a d d i t i o n , d iv F = 0 _in_ Ω and F ·η = 0 _on Γ ™U Γ , we have
0 < 6 0 = min S x ,t) < Sxyt) < max SQ(x , t ) = 1 - δχ < 1 . (4 .6 ) Ω^? Ω™
REMARK. In f a c t , in (4 .4 ) we get the same r e s u l t by t a k i n g the minimum and
maximum only on Γ ~ υ Γ η Γ Π ϋ Ω . J IT IT 0
PROOF. In o rder t o prove the i n e q u a l i t y Θ < a± , we cons ide r the func t ion
Θ def ined in Ω by Q(x , t) = max θ(χ, t ) — α , θ ; Θ i s such t h a t
θ = θ - α 1 on Aa = x 9t) e Q,m : Qx9t) > 04 ,
Θ = 0 and V(J = 0 a . e . on Üm-Aa .
We denote by θ~ the func t ion Θ (x , t) = max d Jx , t) — αχ , θ , where
86 Y. Atik ΐ + δ
Q^(x,t) =j [ θ(χ,σ) do; t
Θ is extended by 0 outside [0,T]. Let τ, 6, and r be positive numbers such that
2 0 < τ <T , x Q = T - o > 0 , a n d 0 < - < T Q . Let η be the function r\(x 9 t) = 0~(a;,£)x (t) , where
Γ 0 ; t < 0 or t > τη
xr(*) = <
o It is the obvious that the function
r* ; *€[0,?] 1 ; t € [i , τ -I]
^(τ -t) ; te[T -i , τ0]
belongs to the space W11 (ΩΤ , Γ^ U Γ^) n Ζ^ (Ω^) and Π - ( χ ' ^ " ° f o r
£ < 0 or t > 1 . Substitution of η^ in (γ.1) leads to o
- ( Ι θ · η δ Λ Τ + ^ · V8>fi + <Λ*νθ νηδ>Ω = ° · T τ τ
where Ω = Ω X ] 0 , τ [ . Letting v -> + °° , we obtain
| ζ(^)[θ6(χ,τ)]2-[θδ(«,0)]2ώ + ( ( ^ · 7 θ ) δ , θ δ ) η τ _ δ + ( ( Λ Λ 7 θ ) δ , ν θ 6 ) β τ _
Thus, by letting 6-> 0 and using properties of Θ, we obtain
= 0
[ ι [ θ ( χ , τ ) ] 2 dx + (A*ve9ve; Ω
« = - ( ν ν θ , θ ) Ω · τ τ
Using the d e f i n i t i o n of ΛΛ , ( i ) , ( 3 . 1 ) , t he l a s t e q u a l i t y , and remembering
the i n e q u a l i t y
|V0| | θ | < | α | ν θ | 2 + ^ | θ | 2 , α > 0 , we get
For a s u f f i c i e n t l y small t he cons t an t 5 (A/-1—r) i s p o s i t i v e , t h i s impl i e s
y ' C O < M ^ C T ) , for almost a l l τ € ] 0 , Τ [ , where M1=M/(ah), and τ
2/(Ό = J J [ 6 ( o ; , t ) ] 2 ^ 6?t. 0 Ω
Because z/(0) = 0 and Gronwal l ' s lemma, ζ/(τ) = 0 a . e . in ] 0 , T [ ; then 6 = 0 a . e . i n Ω„, we conclude θ ( ι , ί ) < a, a . e . in Ω™ .
Heat and Mass Transfer in Porous Media 87 The proof of aQ < Q(x,t) a . e . i n Ω i s c a r r i e d out in t h e same man
ner by c o n s i d e r i n g the f u n c t i o n θ = π ι ί η θ - α , 0 .
To prove (4 .5 ) we cons ide r t h e f u n c t i o n s S = max S — 1 , θ ΐ , § ~ , φ = 5Γ>γ , 1 J 0 0 T
and φτ which is in W ' (Ω , T2m)> Furthermore, φ-(χ 9t) - 0 for £ < 0 and £ > τ . Substituting φ in (γ.2) and going to limits with respect to r and
o δ , we obtain
i J m(x) [S(X,T)]2 dx + (K0aVS + FÄ , VS) Ωτ = - ( d V^ · VS , 5) fi . Ω "τ
Using properties of 5 and observing that F*(% »Θ j5) = Fx , Θ , 1 ) = 0 for 5 > 1 , we get
\ j m(x)[s(xsx)]2 dx + U 0 a V S , VS) Ω τ i
We finish the proof as above. In order to prove (4.6) we observe that
(?.V<P)f Ωπ (div F,tp)0 - (F-η,φ) \Q
therefore, (γ.2) becomes
-(mS,vt)ü + ( Χ 0 α ν 5 , ν Φ ) Ω + (^ 7^ · VS,(p) Ω = (mS0,<p)Q
Now, the proof can be performed as above by considering 5 = maxS-l + δχ , 0 and S = min S-6Q , 0 .
5. Existence of a Generalized Solution of I -i . In order to prove the existence of a generalised solution of system I , we make use of Galerkin's method. Assumptions 3.1 ensure the existence in ΩΧ [0,T] of Galerkin's approximations θ , S , and P satisfying the estimates
ρ * ( · · ' > | ! . 0 * Ι ' ρ " < · · ' ) | 2 , ο < < ; '' ' " 1 0 · ' ' 1 ·
''(•*)U..H"1..0.
Δ θ'
* ^ 2 , « + « V S U2,i
2 , Ω, < C
^ Λ II 2 , Ω I o
0 II 2 ,Ω I o
Δτ„5 lU.ß <σ!το
< C a . e . in ] 0 , T [
<C a . e . in ] 0 , T [
a.e. in Ω^
a.e. in Ω^
a.e. in Ω
a.e. in Ω. L0 " ^ » üby
where C denotes different constants independent on /V and t,
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
88 Y. Atik
Δ QN(x9 t) = QNx+xn , t)-QNx9t) , s ; n e i R 3 , and ^0 °
Δτ QNx9t) = QNx 9 t + TQ)-QN (x9t) , τ 0 € Μ .
The estimates (5.1) —(5.7) permit us to extract from P , θ and
S (we keep the same notations) subsequences such that
PN( · 9t) ^ P( · , t ) , VP^( · , t ) P( · , t ) weakly in L (Ω) a.e. 2X
in ]0,T[
Ϊ7 * e A ? ( x , t ) > Q(x9t) , S ^ ( : c , t ) > Sx9t) a . e . in Ω
θ ^ > Θ 5 ^ > S s t r o n g l y in £2(Ω ) , and
ν θ ^ ^ V0 VSN ^ VS weakly in L2(fi ) .
One can, then, prove that P satisfies (a) and (d.3) of definition 3.3 with
-V=KVP + f . Put / - P - / , ^ = K(x,QN,SN), and f - / ( * , Θ* ,SN) .
Writing
( / v / , ν / ) Ω = ( K W , ν ( ρ - ρ 0 ) ) Ω + ((K"-K)VP, v (P0 - Ρ % +
+ ( ? " - / , V(P0 - P % , using (i), and properties of the above sequences we can in fact see that P^ -> P and VP^ ->VP strongly in L2(Q ). It follows that -VN = KNVPN +fN
Vitali's convergence theorem we have converges strongly in L (Ω ) to —V=K^JP+f. Therefore, using (3.1) and
eorem
^ = ^ / ( 1 + Ä | F | ) — * v = v / \ + h \ v \ )
strongly in (L (Q )) . This last result, assumptions 3.1, and properties of θ , and S imply that Θ and S satisfy identities (γ.1) and (γ.2) of 4.1 (without stars, in virtue of 4.3). It is obvious that P,Q,S satisfy the analogues of (5.1), (5.2), and (5.3); they are then in V (Ω ) and VP € L2 m(9, ). Also Θ,5,Ρ satisfy condition (c) of 3.3 because QN,SN,
Y are chosen such that this condition is fulfilled. The functions Θ and S are bounded in virtue of 4.3. The boundedness
of P can be proved as follows: We consider the problem
-kW = 0 in Ω and | | = g on Γ (J Wdx = \ gdY = 0 j , (5.8)
g[x9t) =\o for a: €Γ , Q(x9t) for x£T , - J ~ J \ Q dT for χ£τΑ ,
here |Γ | > 0 is the measure of Γ .
For Γ smooth this problem has a solution satisfying
(νϊ/,νψ)Ω= (0,ψ)Γ for ψ € ^ ( Ω ) , (5.9)
and
Heat and Mass Transfer in Porous Media 89
^\\W\Q) < const. I gJI for 1 < p < nq/ ( n - 1) = 3 q/2 . (5.10)
to see the estimate (5.10) i t suffices to write the solution of (5.8) in the
form of a single layer potent ia l [3] and use theorem (12, VII) of [8] , p.27.
Adding (5.9) to (d.3) of def ini t ion 3.3 we get
(XVP+ / * , ν ψ ) Ω = 0 , ν ψ Ε ί ^ ( Ω , Γ 2 ) , (5 .11)
where / * = / +Vi^. Then we obtain
Ι?1ρ.-,0Γ<«ί»"·ΡΙρ.-,ΩΓ
+ΙβΙΡ.-,Γ2.· P6]3.3,/2[. Taking in (5.11) ψ = Ρ^ = max ± P- j , 0 with j > M we get
M-2||vp(j) ||* Ω < M-i (XovpU') ^ νρ(^>)Ω < (KVP(J\ ν ρ ( ^ ) Ω .
(XVP + /* , νΡϋ )) Ω= (KVP^^+f* , VP(^) ,
Then
It follows that
A. = (χ£Ω: ± P(ar,t) > j \ , ΰ
(xvp0') > s!Pi))A =-(*, VF(J))A .
V P ^ I 2 , <Af* f if*|2^<M4(mes/l,)(P-2)/P II2 > · J J 5 4 .
^ ,2 J c·*
(p-2)/p „ < const, (mes 4 )
On the other hand, using Poincare's inequality, we have
i < const. (mes/l.)i+1/3|vp(i)| fi. Consequently
[ (±P-3)dx< const. (mes^.)1 + ( P " 3 ) / ( 3 P ) · A . J Applying lemma 5.1 of [5], p.71 with ε=(ρ-3)/(3ρ) and a= 0 , we conclude
that \p(x,t)\ is bounded. Therefore, the existence of a generalized solution of system I , is
proved.
6. THEOREM (Compactness principle) (see [1]). Let S be a sequence of measurable functions in Ω such that
T
* 0<5 £ (x,t) < l a . e . infiy, (6.1)
90 Y. Atik
SC(x,t) = SQx9t) for [x9t) £ T2T, and (6 .3 )
* Vtp € Wl2fl (Ω , VzT) wi th φχ9Τ)=0 for_ a? € Ω , we have
<mS - m S o - « P t ) n r = ( r ' V ( p > V ( 6 · 4 )
here α„(τ) (recall a(x 9Q 9 S) = a AS) P (χ)/μ (Θ)) is a continuous function, positive for 0 < τ < 1 , and
α0(0) = α0(1) = 0 , (6.5) ~ ε where S , m(x), and A are functions satisfying
(l· 0<5Q(a:,t) < 1 a.e. _in Ω^ , (6.6)
I ^ I U . n ^ N o f ^ o L . n i l tomIX«. (6.7) Then from S we can extract a subsequence S %\ such that:
r ε ·ι * \S τ \ converges a.e. in_ Ω and on V and converges strongly m
L itim) and in L (Tm) for 1 < a < + » to a function 5.
* Slu τ = I <ζ0(ξ) ί?ξ converges weakly to Vu Ξ α0(?) ζ IE ^ο^Τ^ *
5 0 PROOF. For the sake of convenience we shall assume A x9t) defined in Ω XIR (for instance A£ x, t) = 0 for t ψ [0,T]),then (6.4) is considered in Ω Χ ϋ . Taking in (6.4) φ( x, t) = ξ(χ) \i(t) with ξ€#*(Ω,Γ ) and μεΡ(Ο,Τ) we deduce that
d (mS^-mSQ, ξ) Ω = - ( , ν ξ ) 0 in Ρ'(Ο,Τ). (6.8)
e two fixe Integrating (6.8) we find Let τ and τ» be two fixed numbers τ > 0 , T_>0 such that [τ , τ + τβ] «=[0 , Τ].
I τ + τ 0
(χ) Δτ (5ε(χ,^)-5(χ,τ)) ξ(χ) d x = - J ( 1 ε , ν ξ ) Ω ^ ^ . (6.9) I 0 Ω
Now consider the function _, ,. S(x , t)
Z(S) = Zx9t) = \ s/a (r) dr 0
which is continuous and strictly increasing with respect to S; for dZ j ^ = VaQ(S) > 0 if S e ] 0 , l [ .
Then Z has an inverse function S = S(Z) which has the same properties. It is easy to see that (Z = Z(S )):
0 < Z e ( x , t ) < M = max Van(S) , II VZ£| < M ςρΓθ 11 " " 2 ,Ω™ ύ ^ [ υ , Μ i \ (6.10)
Ί2 I X ^ ) - ^ ) ] <W2(51-52)[Z(S1)-Z(52)]
Heat and M a s s Transfer in Porous Media
Choosing in (6 .9) ε S (χ,τ)
ζχ) = Δ τ ue = Δτ [ j >ία^) dr] SQ(x9T)
which i s in W (Ω,Γ ) (ξ i s a func t ion of x only because τ and τ a r e
f i xed ) we have
m(x) Δ (S£(X,T) - SΛΧ,Τ)) Δ u£(x,T)dx
o ° T ° Ω τ + τ
- jVe,
91
νΔτ u) dt. Π "
On the o t h e r hand, accord ing to (6 .10) we have
Δ Z(Sb(x,T)) dx < M2M λ f Δ τ 5 ε ( ^ , τ ) Δ τ Z ( £ ε ( χ , τ))άχ ,
but
Then
Μ"1 I |Δ Z[S' Ω ' °
Δ τ 5 ε Δ τ Ζ ( 5 ε ) = Δ_ ( 5 ε ( χ , τ ) - 5 _ ( * , τ ) ) Δ τ ^ ε + Δ Τ 5 η Δ τ Ζ(5 ε ) το το το ° ο το ° το
ί Δ Ζ(5 ε ) dx<M~lM [ Δ ( 5 ε - 5 ) ( χ , τ ) Δ uz dx Ω ' ° ' Ω ° ° 0
+ Μ^Μ 2 ί Δ 9 0 Δ τ Ζ ( 5 ε ) + Λ 5 ε Δ τ Ζ ( 5 ) - Δ S Q A Z ( S Q ) U x . ^ L 0 0 0 0 0 L ϋ J
\mx) Δχ ( 5 ε - 5 ) Δχ ue dx Ω ° °
τ + τ ί | 2 ε d t , νΔχ uC)
The f i r s t term on the r i g h t y i e l d s
M~l Mn [ Δ (Se-S )Δ uZdx<Mn 2 J τ ην o y τ 2
= - Μ^
0 Ώ Changing the variable t = τ + τ σ in the last integral we get
τ+τ ( I Ae(x,t) dt, νΔτ ue)
0 Ώ
If f ^ 2ε(χ,τ+τ0 σ)ίίσ , \7Δχ u* ;Ω
Since
also Δ Z(SC) , Δ 5ε , Δ Ζ(5η) < const., τ0 το τ 0 ° I
Δ Z(S )
the second term on the right yields
2| τ0 0
92 Y. At ik
AT1 M0 f Δ SnL·. ZSE) + Δ S £ Δ Z(S )
Thus
Δ £Λ Δ Z(£ ) dx < c Ω
Δ 5 τπ 0 dx ,
M"1 Δ Z(SE
12,Ω M 2 | T O ! f I Κ ( * > τ + τ ο σ )
Ό Ω '
VA u dx d a + c i l τ0 0| dx ,
where Δ S~ can be w r i t t e n
i 35 Δτ SQ =SQ(X,T + TQ)-SQ(X,T)=TQ \ ^ ( x , T + T Q a ) d o
0 0 Integrating now with respect to τ from 0 to T — x we get
K ^ ) \ \ 2,i 'Τ-τ
Τ-τ l
0 0 ο Ω Τ-το ι, . 3S
< β2 Ι τ ο 1ε(^,τ+τ0 σ) τ0
dx da dx
Writing I9 in the form
f r r I o 1 + .1 J J -5ϊ-(^τ + τ0σ)^^σ^τ|^σ2|τ01^+^
0 0 β
J2 = f do j dx j J U-^- (x, τ + τοσ) dxj , 0 Ω
changing the variable
τ + τ0 σ - t [dt = dx and 0 < τ0 σ < t < Τ + xQ σ (σ - 1) < Τ)
i n the i n t e r i o r i n t e g r a l , and i n t e g r a t i n g over [0,T] i n s t e a d of [ τ 0 σ , T+TQ ( σ - l ) ] we ge t
f f I ^ o I i J2 < J J ΊΓΓ^'^Ί d x dt d° < (mes V * o ςιπ
dsr
dt 2,Ω Γ *
For I. we have
*,</[/ | rc τ + x σ o
dxdT 1* f ΔΤ V ^ T d x dx
%J I L o I
d a
T - T ^ - T n
But
τ'το p u t t i n g t = τ + x a , we find
T-x
T-x
ί Ω 0
Γ 1->ε / I 2 Γ Γ °|->ε I 2
J U ( ϊ , τ + τ σ ) dx dx = j L4 ( χ , χ + χ 0 σ) dx d x ,
f 0 k r 12 ( \+r I 2
J L r ( x , x + xQa) dj < j \AE(x,t)\ dt .
Heat and Mass Transfer in Porous Media
Then (according to (6.7))
93
L Ω 1 , '
I 2 ~P τ + τΑσ) dx άτ\ < III ^ ' \ " ~ ~ \ | *\Γ ΙΙ2,Ω, < M ,
"T~Tv Concerning the integral
ί VA uE dx dx
we have Τ-τΛ
VA u \ <4 V Z b ( x , T + T o ) + VZ ( χ , τ + τ
where Ζε = Ζ(5 £ ) and Ζ0 = Ζ ( 5 η ) , then
Ω
VZ (χ,τ)
f I r Iz VAT w ^ d x < 4 ( M + M9)
VZ (χ,τ 21
Γ-τ0
(by (6.1), (6.2), and (6.7)). Finally, we conclude that
< const. *
Furthermore, using (6.2) it is easy to see that
πε II Ι2,Ω„ < M
(6.11)
(6.12)
where Δ Z (x , t) = Z (x + xa ,t) - Ζε x , t) , and xn e M fixed. ^0 °
To see what happens on the boundary we use the inequality
N L r M H I l J M I ^ + H k J true for a l l u in ,\ü) ,
where a£ [ 0 , 1 [ and q = 2(n-\) / (n-2a) , see [4 ] . Then for a = - we havee
Δ τ , Ζ Ό ι ΐ 2 ' Γ
( a cco rd ing t o (6 .10) and ( 6 . 1 1 ) ) . Thus
< c o n s t . ( | τ I "* + Ml+N* Δ τ Ζ | | 2 ) Γ < c o n s t . | | τ ο Γ + •ΚΜ'.Ι'·
Similarly, taking VA_ Z < C(M) into account, we find that 2,Ω 0 II Z y^Crp
ε ί' Δ^ Ζ 2,Γ7 < const •hol* + |*ο|·
(6.13)
(6.14)
From (6.1) and (6. 1 1) — (6.14) we conclude that Z and Z I p ] are strongly compact in £2(ΩΤ ) a nd ^ 2 ^ 7 ^ respectively. Since the function S = S(Z) is continuous the sequences Se and S | Γ have the same properties. Finally, using this compactness and elementary results of integration and distribution theories, we can extract subsequences satisfying the conclusion of the theorem.
94 γ. Atik 7. Apriori Estimates and Passage to the Limit with Respect to h .
h 7.1 Apriori estimates independent of h. Let (Θ,5,Ρ) = (Θ (x9t) 9 h h
S (x,t)9 P (x,t)) be a generalized solution of system I , . We know from Theorem 4.2 that
α < θ ( α ? , £ ) < α and 0 < £ ( ; E , t ) < l a.e. in QT. (7.1) We are going to use (7.1) to get apriori estimates independent of h. To this end we choose in (4.1) and (4.2) (without stars) the test functions
t η - ( * , * ) = - £ f [9x9o)-QQx9a)]\o)do
6 t-6 and t
φ_(χ,£)=| j [s(x9o)-SQ(x9o)] Xr(o)do
(where χ is the function, which vanishes for £<0 and t> τ-6 , given in the proof of 4.2); we get, by letting r-> + °° and 6—>0:
<ΙΘ* · e - 9 0 ^ + $h ' (θ-θθ)νθ>Ωτ+ (Λνθ,7(θ - 6 0 ) ) ^ = 0
and (^5^,5-50)Ω + (X 0äV5+F, V(S-50))fi + fr^ -VS 9 S-SQ)Q =0.
τ τ τ Making use of (i), (iii), (iv), (7.1), and the inequality
-, a 2 , 1 r 2 aZ? < -75- a + —— b 2 2a with a suitable choice of a > 0 , we deduce from these equalities that:
+H22vll^v · ·
where C is a constant independent of h. I 2 Let us now show that V7 I „ is bounded independently of h. We have
II /z II 2 , Ω/ρ r
7 ^ Ι | 2 , Ω < lr II2,Ω · Then
i M l l . n < IK^IL«< c o n s t · ΙνρΐΓ2,Ω+ IC« · But, for VP = V P ^ we have the estimate
which can be inferred from (4.3) by taking \p = P — P and making mainly use of (i), the properties of traces, and Poincar£fs inequality. This gives for V-, the estimate
..r,
Heat and Mass Transfer in Porous Media 95
where C is a constant independent of h,
Consequently, since F and / are bounded on Ω , we deduce the existence of a constant C9 independent of h, such that
Ι Ι 2 , » , Ω Τ II Ι | 2 , Ω Τ II Ι Ι 7 2 ( Ω Τ ) II \\ν2Ωτ) 1 Λ)
7.2 Passage to the limit with respect to h. Let (Θ , S , P ) be a genera-h h
lized solution of system I -,. The functions Θ and S satisfy (7.1), (7.4), (4.1), (4.2), and condition (c) of 3.3 in which 9Q and SQ verify (iii) and (iv). Furthermore, consider the problem
-kW=mSot in Ω and W=0 on Γ=8Ω, which has a solution in W (Ω,Γ) satisfying
(νί/,νφ)Ω =-(^^ 0> φ ) Ω " ^ Ο ' ^ Ω ' Υ φ € '^T'V
with φ(χ9Τ) = 0 for a l l χ € Ω , (7 .5 )
and
™L,nT^c\\sot\\2,nT. (7.6) h The a d d i t i o n of (7 .5 ) to (4 .2 ) shows t h a t S f u l f i l l s
mSh-mSQ,if>t)ü = ( 2 / 2 , ν φ ) Ω + | ß\dxdt, ν φ € w\91 (Ω^ , Y lT) ϊίψ
with φ(χ,Τ) = 0 for χ Ε Ω , where
A = KQ a V5 + F + W and ß =bqV,»VS .
Using 3.1, 3.2, (7.3), and (7.4) we see that
Similarly, by considering the problem
-M = IQ M in Ω and X = 0 on Γ , h we can show t h a t Θ f u l f i l l s
ZQh -IQQ , η^ ) Ω = ( 2 ^ , ν η ) Ω + ί B^dxdt , for a l l η as in 3 . 3 . d ,
where
with
Since
lh=AhVBh+VX and B* = Ψ. · V6h
1 1 / 2
»T II Η Ι Ι , Ω ^
[ Bhipdx dt and [ £^η Ωτ Ω^
dx dt
96 Y. Atik have good behaviours, we conclude that a nondegenerate version of the compactness principle applies for θ and s . Since |VP satisfies (7.2) it is possible to extract subsequences such that
* Qh[x9 t)-> θ(χ, t) and Sh(x , t) -> S(x , t) as /z -> 0 , a.e. in Ω , so that
aQ < 6(x,t) < αχ and 0<S(a:9t)<l a.e. in Ω^ . (7.7)
* Qh -> Θ and Sh -> S strongly in £2(Ω ) .
* Sh\v -> S\ strongly in L2(T) .
* sjQh -^ V0 and VSh -- VS weakly in £2(Ω ).
* Ρ^ — ^ P and VP^ -> VP strongly in ^(Ω^) . (For this last result, see in section 5 how we prove that F —> P and VP^ —>VP strongly in ^(Ω^).) Therefore
-fh = KhVPh + t->KVP + f = -V, and f1 / (1 + Η\Ψ\ ) -> V
strongly in L (Ω ) as h —> 0. h h
Writing (4.1) and (4.3) for Θ and P and letting h —> 0 we get
- ( Ζ θ , η ^ +(7,ηνθ) Ω +(AV0,Vn)fi = (ίθ 0,η) Ω (7.8)
for a l l η chosen as in (d) of d e f i n i t i o n 3 . 3 , and
( 7 , ν ψ ) Ω = ( 0 , ψ ) Γ for almost a l l t € ] 0 , T [ , ψ € fi£ (Ω, Γ2) , ( 7 .9 )
( then d i v 7 = 0 , 7 · η = 0 on T lT and 7 · η = 0 on Γ T ) . This because θ^ , Sh,
P y K" f A , and f converge s t r o n g l y in £2(Ω ) as /z —> 0 .
Now, u s i n g the formula bH,V<9)a = ( Ζ ) Λ β , φ ) Γ - ( ^ - V S ^ . c p ^ ,
(4.2) may be written for S in the form
-(mSh,vt)u +(K0ah VSh-bh1f+Fh,W)n = (mSQ , «p)fi - (bhQ, Φ) ρ
where φ is a smooth function belonging to ί/ ' (Ω , Γ ) with φ(χ9Τ) = 0 for χΕΩ. For φ smooth we have
\\ SK ' ' ν Ω τ | II £Τ | |ΟΟ,Ω Τ Ι Ι / ζ Ι ΐ2 ,Ω τ Ι Ι | | 2 , Ω Γ
Finally, using properties of sh , sh\T , θ^ , ph , [a1] , ^ and Ϋ1
we can let h-^0 to get
-("ΐ5,ψ^)Ω + ( Χ 0 5 ν 5 - ^ 4 · Ρ , ν ψ ) Ω = (m50 , φ ) Ω - (2?ρ,φ)Γ , (7.10)
Heat and Mass Transfer in Porous Media 97 with φ smooth. Since smooth functions are dense in the subspace of 1,1
W^ (Ω , Γ ) consisting of functions that are equal to zero for t = T, (7.10) holds for all φ€ A/*'1 (Ω^ , Γ ) with φ(χ ,Τ ) = 0 , χ £ Ω .
To finish this section 7 we remark that Θ, S, and P satisfy also condition (6) of 3.3.
8. Apriori Estimates and Passage to the Limit with Respect to ε. 8.1 Apriori estimates independent of ε. Making use of (7.7) we can find apriori estimates for θ = θ (x, t) and S=S (x, t) independent of ε . Using in (7.8) and (7.10) the analogues of test functions r\-(x9t) and ip-(x9t)
6 6 given in 7.1, integrating by parts, and letting r —> +<» and ό —£-0, we get
2 J ι(χ) θ 2 ( ^ , τ ) ^ + (Λνθ,νθ)Ω = -(IQ, QQt) + ΖΘ (a;, τ ) QQx , τ) dx Ω τ 'τ Ω
- \ [ \Q20(x,0)dx+ (Λνθ,νθ0)Ω + ( ν , (Θ0-Θ)νθ) , (8.1)
Ω τ Ωτ
and G = \ J m(x) S2x,T)dx + [KQa V5 , VS)fi = - ( m S , 5 0 ^ Ω
Ω τ τ
IJ m5(o;,T) SQX,T) dx - \ j TTZS* (a:, 0)<fo + (#Q aVS , ν£ 0 ) Ω
Ω Ω ' τ
+ (bV9V(S-S0)\ - ( F , V ( 5 - 5 ) V - ( & ρ , 5 - 5 0 ) Γ . (8.2)
According to (7.9) the term (bV ,V(S — £ Q ) ) 0 can be rewritten as
(^,VS-VS0) =(ö,w) r + [V9b(SQ)-bS))VSQ)Q , T IT ' ' 'T
W h e r e S(x,t) u(x9t) = b(o) do
SQ(x,t)
which s a t i s f i e s Vu = b(S) VS-bSQ ) VSQ , \u\ < II Z? H^ \S-SQ \ < 2 \\ fcfl^ , and
w | r = 0 ; t h i s because 5 = 5 . on Γ . Concerning t h e f u n c t i o n S we have •l 2T7 0 2 i
the estimates:
* ( X 0 a V 5 , V 5 0 ) f i < 1 CMa (X0 aV5 , ν 5 ) Ω + - ^ CM (KQ a VSQ , VSQ)Q ,
* | ( ?>v( s- so))ß | < ί ^ ^ I v s l & d t + i p f ^ K l L n 1 x ' τ ' ^ v a τ τ
τ
98 Y. Atik
b v , n s - s 0 ) \ (Q,u)T
ιτ V ,(b(SQ)-b(S))\!sja j
HMUl^^/HUllCß/IKIlU < C (^ΙΗΙΙΩ HKILn +ΙΗΙΙ(Γ ·
-mSQ, SQt)ü + I ffl50(xj) S ( x , τ) c ? . x - | J mS^(ai , 0 ) dx — (2?« , S SQ)V τ Ω Ω 1 T
WJIkotlli , < £ , Ω + 5 0 ο ο , Ω + \\Q ΐ , Γ τ τ We w i l l r e f e r t o t h e s e i n e q u a l i t i e s by ( 8 . 3 ) . According t o ( 2 . 3 ) , 3 . 2 , and
3.4 we have
|F | /fa < |F | /fa < M\
On the o the r hand II | |2 H ^ ii 2
(8.5)
ν νΑ
ρσ
+ ( Ρ ι - ρ 2 ) ^ <M0(M) (8.4)
and ^''τ€,α+1^422,Ώτ
<0^°
..r.J· (8 .6 ) I 1 2 2 . « r< c ( M ) K l L q r
+ I ? l 2 . « J , + i . - . . ι τ Choosing a small in (8.3) we can infer from (8.2) , (8.3), (8.5), and (8.6) that
I5 II n + Ι Ι^ ν 5 1ο I Ι Ι 2 , ° ° , Ω Τ II II2
II oil 2 , Ω Τ II ο Ι Ι ° ο , Ω Γ II ο | | 2 , Ω Τ < C , (8.7)
where C is a constant independent of ε; this because £', F/v a , and /* are bounded on Ω .
For Θ we deduce easily from (8.1) that II II 2
I W ^ IKJio+ IKI ! , Ω / Ι ν θ ο Ι ' , Ω / Ι ν ρ ο ΐ 2 , Ω Τ
+ Ι ^ 2 , Ω Γ + | Ο 1 2 . Γ 1 Γ | < ^ (8 .8 )
where C is a constant independent of ε.
8.2 Passage to the limit with respect to ε. By considering the problem
Δί/ = mS in Ω , W\ =o., Ot IK dn
which has a solution satisfying
= 0 , and ^- L =bQ, Γ ^n I l i i l 1
(Vtf, Vcp) = - (m5 , <pJ0 - mS ,φ) + bQ,<p) IT
for all φ as in 3.3(d), and
Heat and Mass Transfer in Porous Media 99
hl2^<c o n s t-IsoJ2,n/ IH|2, rJ. (a.« ε one shows t h a t S = S s a t i s f i e s the i d e n t i t y
mSe-mSf], φ , ) 0 = (2£,Vcp) for a l l φ a s in 3 . 3 ( d ) , U V U lim
where T = ^ 0 α ε ν 5 ε -bCVe+VW.
According t o ( 8 . 6 ) , ( 8 . 7 ) , ( 8 . 9 ) , and t h e boundedness of F and b we
have
\r\\^<C(M). (8 .10)
S i m i l a r l y , by c o n s i d e r i n g the problem
-AX=l(x)dQt i n Ω and l L = 0 ,
one can show t h a t Θ f u l f i l l s
( Ι θ £ - £ θ η , η ) = ( 2 ^ , V n ) 0 + flfndxdt fo r a l l η a s in 3 . 3 ( d ) , o
where 2 ε = Λ ε ν θ ε + VZ and £ £ = 7£ · ν θ ε . According t o 3 . 2 , t he analogue of
(7 .3 ) fo r V , the i n e q u a l i t y v*IUr
< c o n s t-lk*l|2 ,v and (8.8) we have / M _>. ιι ιι ιι \
( M f L o > kf i o < const. VII i ΙΙ2,ΩΤ II i II ι,Ωτ;
In addition, S and θε fulfill (7.7) and 3.3(c). Hence the compactness principle applies for S . Thus, we can extract from [S a subsequence ε ·
\S ^ having the properties mentioned in the conclusion of the compactness principle. Since the term *
B η dx dt
have a good behaviour we can apply a nondegenerate version of the compactness ει ε ·, principle to θ° to produce a subsequence θ τ \ which converges a.e. in Ω
and strongly in L (Ω ) for 1 < q < + °° to a function Θ and such that V0 ^ —^V0 weakly in L (Ω-) . Furthermore, since || VP£ P < C it is possi-
2 1 £ £ · * > ύί ble to choose the extracted subsequences s τ\ and θ in such a way that ε ♦ .ε ·
P τ) and V τ] converge strongly in L (Ω ) . Denoting by Θ , S, P, the ε - ε · ε · *-
limit of θ ^ , s , and p z] respectively, it is easy to see that P and Θ satisfy
(KVP + f ,νψ)Ω = - (£,ψ)Γ a.e. in ] 0 , T [, Υψ <Ξ ( Ω , Y2 ) , and
-(ΖΘ ,nt) + (F.nve) + (Λνθ, vn) = (^,η^ ,
100 for all η in wl'l(tim9 Γπ φ U Γοφ) Π L (Ωφ) with η(χ,Τ)=0 for x€tt, where 2Ty Ύ2 ^ Τ ' Γ1Τ 7 - - KVP-~f .
For 5 we rewrite (7.10) as
Y. Atik 2T)
mSE ,tpt) + (κ0αεν^ε- -&ε7ε + ?ε,νφ)0 + ε ( ί ν / , ν φ ) ,
The difficulty here comes from the term ε(#π V Se, \/φ) 0 IT
However, since
e(X0VS ,νψ)
< M /ε" J JE
\vs
\vs
dx dt /ε |vse| |νφ| da; <f£
dx dt< Μ/έΙΙ^ν^ |2,ΩΦ 2,Ω7
< const. vE Φ 2,ί this difficulty can be overcome. Therefore, letting ε —>- 0 we get
-mS^t)ü + K0aVS-bV + F,W)ü =(mSQ9<p)Q -bQ,v)v
1,1 IT
for all φ e W^L(^T,T2T) with ψ(χ,Τ 0 for x£Q.
Using methods already seen one can show that Θ , S, and P are bounded and satisfy conditions (a), (b), and (c) of definition 3.3.
This completes the proof of the existence of a generalized solution of system I.
REFERENCES 1. S.N. Antoncev and V.N. Monahov, Boundary value problems for certain
degenerate equations of continuum mechanics, filtration of immiscible fluids, Lecture Notes, Part 2, Novosibirsk Gos. Univ., Novosibirsk, 1977.
2. J. Bear, Dynamics of Fluids in Porous Media, Amer. Elsevier, N.Y., 1972. 3. E.B. Fabes, M. Jodeit Jr. and N.M. Riviere, Potential techniques for
boundary value problems on C^-domains, Acta Math. 141 (1978) 165-186. 4. O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and
quasilinear equations of parabolic type, Nauka, Moscow, 1967; English translation, Amer. Math. Soc.3 Providence, R.I., 1968.
5. O.A. Ladyzenskaya and N.N. Ural'tseva, Linear and quasilinear elliptic equations, Nauka, Moscow, 1964; English translation, Academic Press, N.Y., 1968.
6. A. Lagarde, Considerations sur le transfert de chaleur en milieu poreux, Revue de l'Institut Fransais du Petrole,(1965), 383-446.
7. J.M. Menot, Drainage et gel des milieux poreux partiellement satures, 'Docteur—Ingenieur' thesis, Ecole Nat. Ponts et Chaussees, Paris, 1979.
8. C. Miranda, Partial Differential Equations of Elliptic Type, second edition, Springer—Verlag, New York, 1970.
9. K. Raznjevic, Handbook of Thermodynamic Tables and Charts, Hemisphere Publishing Corporation, McGraw-Hill Book Company, New York, 1976.
10. Y.C. Yortsos and G.R. Gavales, Analytical modeling of oil recovery by steam injection: Part I -Upper Bounds, S.P.E.J. (1981), 162-178.
REPRESENTATION OF MULTIPLIERS OF EIGEN AND JOINT FUNCTION EXPANSIONS OF NONLOCAL SPECTRAL PROBLEMS FOR FIRST AND SECOND ORDER DIFFERENTIAL
OPERATORS
N. S. BOZHINOV Institute of Mathematics with Computer Centre,
Bulgarian Academy of Sciences, P.O. Box 373, 1090 Sofia, Bulgaria
1. Prel iminaries. This paper generalizes and simplifies some of the
author 's resu l t s ( [6] , [7] ) . I t s aim is to find some representation formulae
for the mul t ip l iers and mult ipl ier sequences of eigen and jo in t function
expansions of the nonlocal problem.
y' = \y , N(y) = o , ( i) where NEC [a,b] is a nontrivial continuous linear functional in C[a9b] ,
and of the nonlocal spectral problem
y" -q(t) y = Xy , aQy(0) + bQ y1 (0) = 0 , Fy') + G(y) = 0 , (2) where
F9Gec*[o,T]9 t e [ o , T ] , qe^lo.T], | a 0 | + | £ 0 | * o . Now new notions (introduced in [13]) for coefficient multipliers and
multiplier sequences of eigen and joint function expansions are used, in contrast to the same notions used in [6], [7] which are more special than these used now. According to the results in [13] it seems more naturally the "old notions" to be called scalar coefficient multipliers and scalar multi
plier sequences. Now we simplify their representation formula given in [6], [7j and obtain representation formulas for the new more general coefficient multipliers and multiplier sequences as well.
First we recall some abstract notions introduced in [13]. Let X be a linear topological space (l.t.s.). Let L: X —> X, ^r^X
be a linear operator in X. By p(L) is denoted the resolvent set of L,
by σ (L) its point spectrum and let p (L) = XEp(L): X = (L-\l) (Xj-) . Let A C Ö (L) be a set of simple eigenvalues (i.e. dim Ker (L—λΙ) = 1 for λ€Λ and let the root subspaces ILaXT be finite dimensional for all λ€Λ
101
102 N. S. Bozhinov n\ (i.e. now H = ker (L — XI) with ηΛ = dim//, < °°) . Let there exist also an λ λ λ
orthogonal and total system P,, f , of continuous projections P, mapping X
onto // and commuting with L (i.e. P^fP.f, = 0 if X'^ X" and P f=0 for all λ Ε Λ implies /*=0). Then for each fEX we may associate its formal
Fourier root vector expansion (f.F.e.).
It is clear that the injective mapping F:X-> JJ H
XEA λ
is in general a nonsurjective one. We shall denote this correspondence in the usual way: >r-
f~ l pxf-The f.F.e. F is called simple iff dim//, =1 for all λ € Α . A root vector (eigen and joint vector) system v , . . . , V _-^ in H. is called normal
X i f f LvQ = XvQ and Lf, = Xv, + ^ t . - l , 1 < fe < n, — 1 i f dim #, > 1 . We use the n o t a t i o n
x = T T <c x λ€Α χ
and denote its elements by small greek letters: a = ou : 0 < k < n, — 1, λ Ε Λ calling them "sequences"; X is considered with the usual Tychonoff topology. By H is denoted the set of root vector systems in X of the form
U = uk : 0 < k <ηχ-\, X £ h
where τλ λ ι A
is a normal system in //, for each λ £ Λ (we say that U is a normal root Vector system in X). If
;λ λ , K •••••Μηλ-ιϊ is a normal basis in Ηλ for each λ € Λ we say that U is a normal basis root A
sector system in X. It is clear that H is a linear space with respect to r i r ^ i the operations U + 1/ = \U-, + V T \ , aU = j a w j for a € (C .
DEFINITION 1. Inner Cauchy convolution yielded by the f.F.e. F of the operator L is said to be the operation *u: X X X —> X defined by the equality k
λ ^ = 0
for α=ατ,, 3= 3T, EX. Outer Cauchy convolution yielded by F is said to be the operation * : X X H —> H defined by the equality
Multipliers of Eigen and Joint Function Expansions 103
a * n ü - ] ) oL . u . : 0 < k < ηΛ - 1 , \£k\ n [ L k-τ ^ λ ' J J V λ λ
i = o for a = a^ €X , U= uk £ H .
These o p e r a t i o n s g e n e r a l i z e the c o o r d i n a t e w i s e m u l t i p l i c a t i o n for the
simple f . F . e . For t h e i r p r o p e r t i e s see [13 ] . We note only the fo l l owing :
Let LI £ H be an a r b i t r a r y f ixed normal b a s i s system in X. Then a system
\I£H i f f t h e r e e x i s t s a£K such t h a t l/= a *„(./. A system 1/ £ H i s a normal n
basis system iff \J = α *,. U with *„-invertible a £ X ; now ij = 3 Ä (/ with 3 e X , a >νχ ß - ε d i f 1 , 0 , . . . , 0 λ Λ .
Further let Ü = u-,ι 0 < k < n,— 1 , λ € A be an arbitrary normal basis root vector system, and let
V = V i ( / ) " » + " ' + ' l ( / ) V λ Λ
be the representation of the projection ΡΛ with respect to the normal basis λ λ u , . . . , u in // for each λ £ Λ. Let us consider the mapping Λ: J->X
defined by tne correspondence / e ^ / = c o ( / ) , . . . i v l ( / ) i a e x .
I t i s not d i f f i c u l t t o see t h a t A=JU° F where Ju i s an a l g e b r a i c a l and
topological isomorphism between TJ H and X = 7T C λ λ<ΞΛ λ<ΞΛ
depending on Ü G H . The mapping A : X —> X is called Fourier transformation
yielded by the operator L with respect to its normal basis root vector system U£H; f is called Fourier transform of f €. X . The abstract Fourier transformation A : X —> X of L with respect to a normal basis system Ü £ H
is a continuous isomorphic embedding of the l.t.s. X into the space of "sequences"
-r-T- n\
x = ΤΊ c λ . λ<ΞΛ
Although the transformation Λ depends on Ü E H it is more convenient to be used than F because there is a simple connection between all such transformations. It happens that if ~ : / £ X —> f £ X is another Fourier transformation with respect to the another normal basis system 1/6H, then we have ? = 3 * x / , f=a*Kf with α , β € Χ , α * χ 3 = ε , I' = <**„ U , U - « *Hl/. DEFINITION 2. Let Y,Z c l be linear subspaces of X and let #, er Y for each λ € Λ. . An operator A/: 7 —> Z is said to be ( Y, Z ) -coefficient multi
plier of the f.F.e.
of L iff there exists a normal basis system LI £ H with corresponding
104 N. S. Bozhinov
Four i e r t r a n s f o r m a t i o n Λ : Χ ^ Χ and a "sequence" y €X such t h a t
Mf)A=V*Kf for each f£Y. (3)
A "sequence" μ EX is said to be a (Y, Z) -multiplier sequence of F iff there exists a normal basis system U £ H with Fourier transformation A:X-^>K
such that for each f£Y there is / μ £ Ζ with
( f y ) A = y * x f · (3 1 )
For the properties of these notions see [13]. We note only the easily proved identity (Y,Z) = \M £ L(Y, Z) : ΜΡΛ = ΡΛ M in Y and ML = LM in #Λ cm λ λ Λ for all λ G Λ . Throughout (I, Z) and (Y,Z) denote the linear spaces J ö ' c m ms r
of all (Y, Z)-coefficient multipliers and (Y, Z)-multiplier sequences respectively.
An operator M : Y -> Z is said to be (I, Z) -sea lav coefficient multiplier
of F iff there exists a scalar sequence μ = yQk ^ Λ such that P.Mf = yQ ΡΛ /* for each / £ Y and all λ € Λ. A scalar sequence μ = p0k ^ « is said to be (Y, Z )-scalar multiplier sequence of P iff for each f G 7 there is f EZ
such that ΡΛ f = μ Λ Ρ Λ / f°r each λ€Α. The connection between the last notions and the previous notions is evident. The corresponding spaces are denoted by (Y,Z) and (Y, Z)
^ scm sms Now let *:XXX —> I be a bilinear, commutative and associative opera
tion. An operator Mi Y —> Z is said to be a (Y, Z)-multiplier of the operation * iff the equality Mf*g = f*Mg holds for all f,g£Y . The space of (Y, Z)-multipliers of * is denoted by (Y,Z)^. The connection between all previous notions is studied in ([13], [8], [14], [ll]).
Further in both sections a general approach developed by the author in [8], [)!] and [14] is systematically used. 1 k We use the usual notations for the function spaces L , C, C , AC, BV etc. considered on the segment [a,b] in Section 1 and on the segment [0,1] in Section 2. Especially S^norm denotes the space fEBV: f(t-0) = f(t),
BV1 = f£AC : f' €BV , AC1 = fEAC : f1 £AC.
2. Multipliers Connected with First Order Differential Operators. In this section all considerations are made on the segment [a,b]. It is sufficient to solve the multiplier representation problem only for the differentiation D = d/dt instead of the general nonsingular operator fl(x)d/dx + f2(x) because the last transforms to the first by change of variables. Throughout this section we suppose that N£C is arbitrary fixed nontrivial continuous linear functional in C.
Multipliers of Eigen and Joint Function Expansions 105
DEFINITION 3. The formal Divichlet expansion of a function fEL1 is said to
be the formal Fourier expansion (in the sense of Section 1) CO
f~ I Pkf W k=i
on the eigen and joint functions (root functions) of the nonlocal spectral problem (1) (i.e. of the operator D = d/dt considered in X = L1 with domain X = AC = f£AC:N(f)=0). Here Pfe, fc = 1 , 2 , . . . are the Leont'ev projections
V = 2? i i N,\ fu)&Mx~U) du)wi)dx' feLl> (5)
Γ* ° where Γ7 is a closed contour enclosing only \ y of the zeros of the entire def \t function E\) = N+e ). It is known that the spectrum of the problem (1) coincides with the set Λ=λ£,^=1 of all zeros of E(\) . By n, we denote the multiplicity of the zero λ^. It is known [17] that E(\) has either infinite number of zeros or E(X) has no zeros. Throughout this section we suppose that N£C* is such that the formal Dirichlet expansion exists, i.e. the projection system Ρ-Λ-τ._λ is total. We note that the problem for the convergence of the series (4) is studied by many authors, e.g. Schwartz [19] [18], Leont'ev [16], Sedlezkij [20] etc.
Further we note that every λ € (C with Ε(λ)Φ0 belongs to the resolvent set of the operator D=d/dt (X=L1, Xp= Α0„) , For each k the projection Pp maps Ll on the root space
H-, = span and the system . , , ,
Ke , . . . ,t e I [nk- 1) ! |> forms a normal system in H-, , i.e. the series (4) is a series on Dirichlet
±te ek * °<^<^-1
X7t lk
polynomials p-,t) e k , where p. is a polynomial with 0 < deg ρ,^η^-1 . If E(X) has simple zeros only (i.e. if n« = 1 , fe = 1 , 2 , . . . ) then H,= span ß is one-dimensional and the expansion (4) is a simple Dirichlet expansion of the form
L Ak^" k=i
f~ ) Ap(f)ek
As in [6] the basic tool in our considerations is a convolution f*NQ introduced by Berg [l] and Dimovski [15]:
x
f*Ng = N \ \ f(x + t-u) g(u)du\ , f9g£Ll . (6) t
Its properties are studied by the author in [3] . A basic property of Berg--Dimovski convolution is the representation
106 N. S. Bozhinov
Rxf = l-eU/E(X)\ *Nf , f£Ll , A€(C with E(X) Φ 0 (7)
of the resolvent #, of the operator Z), so Z) is an operator with convolu-
tzonal resolvent in the sense of [14]. From the general approach developed in [14] for this class of operators, the following theorem follows:
THEOREM 1. Let X-, be a zero of the entire function E(X) with multiplicity ny Let Γ, be a closed contour enclosing only X, of the zeros of E(X) .
Then: (a) The Leont'ev projection l\ defined by (5) is the unique continuous projection mapping L on H-, and commuting with D.
It is convolutionally represented in the form
where Xt 1
N^k
nn
_ r ±L dx - i ^k
2.Ϊ J 2?<λ) (nf c-D! V 1
( λ - λ λ ) k e U
E(X) X = X-,
belongs to #, and the relation φ-, * φ, = φ. holds. The functions
u (t) = — -sv 2π^ J EX)
Tk
Xt
dx--dx1-
"(λ-λρ^ eXt-E(X)
(8)
(9)
(10) λ = λ7
0 < s < ηΊ — 1 form a "good" normal basis in H-, in the sense of [3], i.e.
k k p N q <
0 for p + q < Wi
fe 0 < p , q < n , - 1 ( 1 0
u , for p + a > n, — \ n - 1 - p - a A: fe fe fe n / > ~ s> \ ·
we note t h a t φ^ = u . , w^ = (#— λ , ) φ, ) k
(b) If
is the Taylor expansion of £"(λ) around λ, , then the relations
1 ,s λ 7,t — t e K / a s - l U l t ) > 1Β^= ), K-IU
1 V 1 = 0
0 < s < n7 1 hold.
(12)
Here 3 : 0 < s < nh 1 are. the first n1 coefficients n k
and the of the Taylor expansion of the function (λ—λ,) k/E(X) around λ7 a relations α ß - I and α^ βν +. . .+ αß'' = 0, 1 < s < ηΊ - 1 if ηΊ > 1 hold.
ϋ ο ο ,9 β υ Α: κ The evident relations
Multipliers of Eigen and Joint Function Expansions 107
k f s 2 π ^ J
E(X)d\ i a° + s +1 *! ^
rJ ( λ - λ / *
2π^ J £(λ) s ! axs
nk-s-l
τι ' Ε(λ) [k
hold as w e l l . Also
E(\) λ = λ7,
0 < k < n . - l (13)
Β(λ) λ = λ , 0<k<nk-l (13·)
λ , * λ ,£ —rt^eK *N — tH e K = \ p ! N q\
0 for p + <7 < n ^ - 1
At η £U)e
< λ -ν λ = λ-κ for ρ 4- q > η* - 1
0 for p + q < η,— 1
) α 7 —Γ t e for p + q > n-, —\
s = o
where I = p + q — n-, + \ h o l d s .
(c) The r e p r e s e n t a t i o n s
V 1 ?1-ι-ι^4^ = l ΐ-ι^ΤΓ*1«* (!4)
of Py with respect to the corresponding normal bases in E hold, where
Cks(f)=*x\ fM jx-u) p k
ΧΛχ-u )
ί(/) = Nx J /(") **(*-")<*"
lu\
, 0 < s < n7 - 1
(15)
(15')
Cks(f) = X a*-t4f) ' Aksf) = Σ 3 β - ΐ 4 ( Λ ' 0 < β < ν - 1 . (16)
Z = o Z = o
The r e l a t i o n s
Z = o 0 < s < nk - 1 (17)
Z=0 j=0 <^*v^= λ <-l L Al-jWK-M
hold for all f,g£L . The proof follows directly from the general Theorems 1,2 in [14]. Indeed
108 N. S. Bozhinov it is easy to see that in the present case all strong derivatives (with respect to the variable λ) in the referred theorems coincide with the usual partial derivatives with respect to the variable λ.
The next theorem also follows immediately from more general theorems in [8] and [14].
THEOREM 2. Le_t 7, Z cz Ll be linear subspaces of L1 and let H.czY, k= \,2,..
Then (l,Z) = (Y,Z). . In particular, the last equality is true if IjZcL1
cm *N __ __ are Frechet spaces topologically embedded in L and if e EY for all λ G (C.
Now we transform some results of Theorem 1 in accordance to the general terminology given in Section 1. Now
oo nh x = TT r k=i
is provided with the inner Cauchy convolution s
for
εν = 1 ij^·«'"!-'!,.,
<-tt· -":» The "good" Dirichlet transformation is said to be the concrete realization of the abstract Fourier transformation
f € L i -*/=c*(/),..., c»ri(/)fe=1ex
pod" normal basis system
«-«^),....«^)^ der also the "usual" Dirichlet
eL^f = Akuf),...,Ak
nk_lf)~^x
lk which is yielded by the "good" normal basis system
defined by (10). We consider also the "usual" Dirichlet transformation
k yielded by the usual normal basis exponent system
f lit n7 -l λ71 ·) V = \e k ,.·.,* k e k /(nfc-l)!J
but which is not so "good" as Ü according to (11'). Further, using (12), (16), (17), we get the following relations
/ = α * χ / , / = 3 * χ / , fiL1 (18)
ü = 3 * H l / , l / = a * H Ü (19)
(f*Ngf=f*K9> (f*N9)~ = α * χ ? * χ ί 7 > f , g Z L l9 (20)
where a <L v : 0 < s < n7 - 1 I , 3 = i 3 : 0 < s < n 7 - W
I s k ik=i i s k k = l
are defined by (13), (13') and - is the other Cauchy convolution in the
Multipliers of Eigen and Joint Function Expansions 109 present case.
In [11], [3], [5] we showed that by certain assumptions there exists a convolutional isomorphism between a certain space jVfczL1 and the multiplier space (Y, Z) Ä . The isomorphism is of the form
J: w€M - > J(m) e ( Y , Z ) * ^ where J(m)f = m*Nf9 fEYfEX (21) or
Ix:m£M-> Ix(m)e(Y9Z)^ where Ιχ(πι) = ϋχ(πι *Nf) , fEY (21 ·)
with DX = D-XI and λ € (C with Ε(λ)±0 is fixed Further from the general Theorem 4 ir. [13] and Theorems 1 , 2 we obtain
representation theorems for the multiplier sequences of the Dirichlet expansions.
THEOREM 3. Let M , Y, Z cz Ll be Frechet spaces topologically embedded in L1
and let Η,αγ, k= 1,2,... . Then: (a) If M s T (Y,Z)*.7 where J is of the form (21), then (J,Z) e = M , i.e. a sequence
μ 4μ : 0 < s < ηΊ - \ \
belongs to (j,Z) iff it is represented in the form k f Γ Ο-?;^ λ. (a:-w) i
p^ = il/J mu) KX y e K du> , 0 < s < nfe-1 , k= 1,2,... (22) o
with m € M . (b) If Μ ^ Γ (Υ,Ζ). where I, is of the form (2Γ), then (Y, Z) = ZX*VM
j. , xy Λ ms A A where
ολ = Κ-λ,1.0,...,θ"=ι€Χ. A sequence μ belongs to (J,Z) iff it is represented in the form
x vl = Νχ\ m(u)Bk sx-u) du\ , 0 < β < nfc - 1 , fc=l,2,...' (22f)
0 with m = H where
ΒΊ (£) = d/dt-X) ·—- e * , 0 < e < n 7 - l .
THEOREM 4. Let M, 7, Z cz L1 be Frechet spaces topologically embedded in L1
and let H^czY for each fe = 1,2,... . Then:
(a) Let M = T(Y, Z) , where J is of the form (21). Then an operator M
belongs to (Y, Z) iff Mf = m * f > f£Y with ??? € M satisfying the conditions
f f q XP(X~U) 1 Α/χ j J m(w)(a?-w) e * d w | = 0 , 1 < s < wfe - 1 (23)
1 0 for each k = 1 , 2 , . . . wi th n., > 1 . A sequence
110
i f f
N. S. Bozhinov
e ( r , z )
c r λΛχ-u) Nx\\ m ^ e d u ) > ( 2 3 ' )
w i t h 777 € M s a t i s f y i n g ( 2 3 ) f o r e a c h k = 1 , 2 , . . . w i t h n-u> 1
( b ) L e t M = r ( 7 , Z ) . w h e r e _Z\ i s of t h e fo rm (21 f ) . Then a n o p e r a t o r M 1X . *N Λ
b e l o n g s t o ( 7 , Z) i f f Mf = Ζλ (m * f) , f GY w i t h m € M s a t i s f y i n g t h e c o n -d i t i o n s r
N \ \ m(u) B. (x-u) du\ = 0 , l < s < n , — 0
f o r e a c h k = 1 , 2 , .
A s e q u e n c e
(24)
w i t h riy > 1 ;
8-1 λ,.ί
( s - D !
" KLi b e l o n g s t o ( 7 , Z) i f f - sms
f x Aj , ( a r -w)
'» 1 , 2 , . . . (24 ' )
with m£M satisfying the conditions (24) for each k = 1,2,... with n > 1. For many cases of isomorphic representations of the forms (21), (21 T)
see [11], [3] , [5] , [10] , [ 12] . We formulate some of them in tables. In the next tables some cases of M to be isomorphic to the multiplier
space (7, Z)+ (by means of J or T, ) for some spaces 7,Z,M are given. N A
The assumptions for N £ C* to allow this are also given.
(a) N£C is arbitrary nontrivial functional
7
BV
BV
AC
AC
Z
BV
AC
Ll
M
BV norm L1
AC
AC
Isomorphism
h h
7
Ck
Ck
BVk
ACk
Z
ck
ck
BVk
ACk
M
Ck
ck
BVk
ACk
Isomorphism
h h
(b) N£C* is of the form N(f)=j fdn where n £ B n o r m has at least one point of discontinuity in [a,b].
Y
L1
Ll
Z
L1
I? BV
M
OTnorm τΡ
Isomorphism
h J
J
1 γ Z
AC
ACN
M
BV norm BIWm
Isomorphism
J
J
Multipliers of Eigen and Joint Function Expansions 111
(c) N£C* i s of the form N(f) = kQf (tn) + / fh or of the form b a
Nf)=\fa) + k2fb)+S fh wi th h£BV, kQ*09 \ k1 | + | k2 | Φ 0 .
Isomorphism M Isomorphism
C
L S^norm
oo L B 7 n n r m
1/p 4- 1/a = 1
(d) N£C* i s of the form N[f) = f fh w i th hEBV. a
Y Z M Isomorphism 7 Z Isomorphism
In the tables we also use the notations: CN = f€C : N(f) = 0 ; C N: ck:N(f) = ...=N(f(k 1 ) = 0 , \<k<< kN r fc
^ r = f ecK y(k)
BVk = f E ACk~l : f(k) <EBV] , 1 < fc < «> .
3. Multipliers Connected with Second Order Differential Operators. We may consider without a loss of generality the Sturm-Liouville operator D = d2/dt2—q(t) , q £ L1 in the interval [0,1] instead of the general linear second order differential operator because both operators are connected with a change of variables.
Let us introduce the nontrivial functionals X0(/) = aQf0) + b0f'(0) , X(f) = F(f') +G(f)t
where F, G £ C* [ 0 , 1 ] . We consider the operator D in the space X =L with domain XD = f £ AC1 ι χ (/) = χ(/) = θ . Let y (λ , t) be a solution of the problem Dy = Xy , y (λ , 0 ) = bQ , y '(λ , 0) = -aQ defined by
y(X9t) = bQyl(X,t)—aQy2(X,t) y
where y (X , t) , y9(X,t) are the fundamental systems of the equation Dy = Xy
(i.e. ζ/^λ,Ο) = 1, ζ/ι/(λ,0) = 0; zy 2 (λ,Ο) = 0, ζ/£(λ,0) = 1). Let us introduce the entire function ^(λ) = x jzy (λ , t) and let
Λ = KLl be the set of its zeros. By n-, we denote the multiplicity of the zero λ , . It is known that the spectrum of the generalized Sturm-Liouville problem (2) coincides with the set Λ.
112 N. S. Bozhinov DEFINITION 4. The generalized Sturm-Liouville expansion of a function fdL1
is said to be the formal Fourier root vector expansion oo
/-£ v where v — 2 b i v d x · k='.2>··· <25> k=1 . Ffe
are the Riesz projections defined by the resolvent R-, of the operator D.
Throughout this section we assume that the functionals χ , χ are such that oo 0
the projection system P^^,= 1 is total, and that Γ^ is a closed contour enclosing only λτ, of the zeros of Ε(λ) . The projection P* projects L
into the root subspace #* spanned by the eigen and joint functions (root functions) of the problem (2) corresponding to X-, .
Further let R-χ be the initial resolvent of the intial Sturm-Liouville operator P, = d Idt —q(t) considered in X = L with domain
λ
X ( 0 ) = ftAC1 : / ( 0 ) = f ' ( 0 ) = θ | . D
It is clear that Ftxf = Rl0)f-X(R(
x0)f)y(\,t)/E(X), f£Ll (26)
and that the following representation of P-, holds, since R, f is an entire function of λ:
, r x(i?i0)f) y\,t)
V"!^] EM < " . * = > > * . · · · (27) In [7] we used a convolution * found by Dimovski and Bozhinov which
represents convolutionally the resolvent P, by y (λ ,t)/E(X). For its properties see [2], [4] . For our aims now it is more convenient to use this convolution multiplied by (—1) which we shall denote also by *. So some differences due to the sign ( —) appear. Comparing with [2], e.g. now we have
Rxf= -2/(λ,*)/Ε(λ)*/ . f ^ 1 · (28)
Following the scheme of Section 1 with help of the general results in [8] , [11], [ 13], [ 14] we obtain immediately analogous statements:
THEOREM 5. Let λ, be a zero of the entire function Ρ(λ) with multiplicity ην ' Let I\ be a closed contour enclosing only λ^ of the zeros of E(X) .
Then: (a) The projection P-, defined by (27) is the unique continuous projection mapping L on H-^ and commuting with D.
It is convolutionally represented in the form pfc/=/*«Pfe. f " 1 (29)
where
Multipliers of E igen and Joint Funct ion Expans ions 113
l f #(λ, ^k i^i J Ε(χ
t) n7 -l ^ k
E(X) dX
(nk-iy. 3 » f c - i
( λ - λ Λ ) feJ/(A,t) ~ W) (30)
λ = λ, belongs to H, and the relation φ, * ιρ, = φ, holds. The functions
2™rJ *(λ) s! 3λ
(X-Afe) * y ( X 9 t )
E(\) (31)
λ = λ7 0 < s <n-,-\ form a "good" normal basis in H-, in the sense of [13], i.e. now (11) also holds. (b) If oo
EM = (λ-λ^) k 7 α\(Χ-Χτ)1
1=0 is the Taylor expansion of E(X) around Xy , then the relations
Ö O η
-h^k^=l %-Λ^> #*>- Σ ί-i TTTä^h-v (32) 9 λ l=o l=o 3 >
0 < s <nv ~ 1 ho ld . fc
Here β : 0 <, s ^ n, — 1 are the first n, coefficients of the Taylor expansion of the function (λ — λ,) k./E(X) around X-,. Now the relations (13), (13f) also hold. Also
1 aP rfv*)*^^.*) Pl ix? dXH
for p 4- q > ft,— 1,
λ=λ7.
0 for p + ς < nv — 1
Z J L a l s ΖΪ 77Z yXk> t] for p + q > nk - 1
;=0 3 λ
where Z = p + g - n, + 1 holds.
nk-i I
(c) The representations
Z = o * Z = o * dA
k i £f_ Pj. with respect to the corresponding normal bases \u : 0 < s < n ^ — l , f i P)S Ί <— - 2 — 2/(λ. , ί ) : 0 < 8 < η , - 1 in ^ hold, where
oX MAA—I
(33)
114 N. S. Bozhinov
•xK(0V
/ ( f ) = -J_- 5 λ dx
!λ = λ„
EM (X-Xk)nkxRWf]
EM X = X-, 0<s<rc7 — 1 ,
(34)
(35)
These coefficient functionals are connected with the same relations (16) and (17) holds also true if *„ is replaced by *.
Further, Theorem 2 holds also true in the present case if *^ is re-Xt
placed by * and e by the function y(X9t). Using the coefficient functionals (34), (35), we can also introduce the "good" and the "usual" Sturm--Liouville transformation Λ and ~ with respect to the "good" system
U = \uk : 0 <s <ηΊ- 1 j°° and the "usual" eigen and joint function basis system
' - & 3λ° /(λ,,ΐ) : 0 < s «n,
k=l
THEOREM 6. Let M , Y, Z a Ll be Freche t spaces t o p o l o g i c a l l y embedded in L1
and l e t H^Y, fe=l,2,... .
(a) I f Μ ^ ΐ Υ , Ζ ^ where J(m)f = m*f, f € Y, w€M , then ( 1 ^ ) ^ = M ,
i . e . a sequence
μ-μ* : 0 < a < V l N
belongs to (Y,Z) iff it is represented in the form
k_ 1 f M s 2Tri J
xR (o) dX
& = 1,2,... with ffiEM.
! dXs Xk J < s<nk- 1 , (36)
(b) _If_ M =Σ (Y,Z)A where I 0n)f = O^im * f) , /€ J, m € M (v £ (C with £(v)^0 vr. is arbitrary fixed) then ( Y, Z) = Z? * M where
:K-v·'·0 ° L e x · A sequence y belongs to (Y,Z) iff it is represented in the forn
^ [ (λ-ν)χΜ°Μ
4 (λ-^)' 0 < 8 < « , - 1 , fc=l,2,... with w E M .
X«Xfc. (37)
Multipliers of Eigen and Joint Function Expansions 115
THEOREM 7. Let M,Y , Z c L be F reche t spaces t o p o l o g i c a l l y embedded in L
and l e t H, c J fo r each k= 1 , 2 , . . .
(a) _Let M= ( J , Z ) ^ where J(m)f = m*f,feY, m € M , then an o p e r a t o r M
belongs t o ( Y, Z ) i f f Mf=m*f, f£Y wi th m € M s a t i s f y i n g the c o n d i
t i o n s
- ^ x f e 0 ) 4 = °> ] <s<n - \ ( 3 8 )
for each Zc = 1 , 2 , . . . wi th w* > 1. A sequence
iff μ0 = x Ä X 0 ) W j ' ^ = 1 , 2 , . . . with mEM (39)
satisfying the conditions (38) for each k = 1,2,... with n-i> 1 .
(b) Let M - j (Y,Z)Ä where Iv(m)f = Dv(m*f), f€Y, m€M (v G <C with Ε(\))Φ0 is arbitrary fixed), then an operator M belongs to (^»^) iff Mf=D(m*f), f£Y with m£M satisfying the conditions v
9λ° L ^ - JJ λ = λ^ 0 , 1 < s < nk - 1 (40)
for each k = 1,2,... with n* > 1 . A sequence
[<λ-ν)χ*ί°4
fc>1-μ = Lk)°° e(y9z)
I 0 j ^ = 1 'sms
(λ^-ν) x | ^ 0 ) w ] , k=\,29... with weM (4i) iff
k μο
satisfying the conditions (40) for each k=\92,... with n*>\.
Another form of the general representation formulas. Another form of previous representation formulas can be found by a convolutional representation of the initial resolvent i?> . Indeed the domain X (Q) c a n be represented in the form X (0) = fZAC1 : xQ(f) = χ(°(/) = θ where xQ(f) = aof(0) + b0f'(0) and χ(0) (f) = -f (0) if bQ * 0 and =-f' (0) if 2>0 = 0. Accor-ding to mentioned result of Dimovski and Bozhinov there is a convolution * representing R^ by the function - y (λ , t)/EQ (λ) where £Q (λ= χ£0^ ζ/(λ , t) . But since y\ , t) = b y (λ, t) —α ζ/2(λ , t) we get #0(λ)=—ι in both cases for 2?0 . Hence
ÄJ[0V= ί/(λ,*)*0 f . feL1, U t . (42) Thus we obtain new forms of the formulas (27), (34) —(41) replacing
χΚ(0)/· in them by xy(X , t) *0 f , e.g.
<Λ-χ'*ο1Π·£»<λ*·*> (43)
116 N. S. Bozhinov
/ ( f ) =x\f *n «:(*)>· (44) (f) =xf *0 /(*)
(45)
r ϊ μ8 = Ψ * ο Β ^ β ^ where B f e , e ^ = D v 7 r ^ ^ · ^ · (Λ6)
T/ze <?ase o/ classical Sturm-Liouville expansion. Now let us consider the classical Sturm-Liouville boundary value conditions χ0(f) =a f(0) + b f7(0) , χ(/) = a,/(1 ) + 2?./'(1) . Many of the previous formulas now can be simplified by means of the pseudoinner product
(f>g) = fu)g(u)du 0
defined for these f9g£L with fg£L. Let z(X 9t) be the solution of the problem Dz = Xz , z (X , \ ) = — b\<> 2/(λ9ΐ)=α1. It is easy to see that x(s) = 0 , XQ(z)=EX), and that x ^ 0 ) /=(/> a ( λ , t)) + E\) KX) where KX) is an entire function of λ. This allows all formulas (27), (34) —(41) to be simplified, e.g.
ί<Λ-<· TT^V*» (47)
k , ^ / * I 3S
< ^ = U · ΪΓ 3λ°
( λ - λ . ) * β ( λ , * )
2?(λ)
(48)
ΐ - 4 ·^^" ( λ * ·*» (49)
i = (m'^,s) where x fc,8(*)=ßvir^s^·*)· (50)
0 < s < n-, — 1 . However it is well-known that for complex-valued q € L the entire function E(X) corresponding to the classical boundary value condition x(f) = cclf(\) + bf,(\) has at most finite number of multiple zeros, i.e. now n, > 1 only for finite k. This shows that formulas (47)—(50) are more simple except for a finite number of k.
REMARK, Comparing formulas (43)—(46) with the corresponding formulas (15), (15'), (22), (22'), in Section 1 for D = d/dt we see that they are very similar. This is due to the fact that the initial resolvent i?5 ' of the operator
Z)(ü) = dl dt x=L , X (0) = f£AC : NQf = - f(0) = θ)
is represented in the form
Multipliers of Eigen and Joint Function Expansions 117
#Λ f = \e *N f by means of the i n i t i a l convolution 0 t
f *N 9 = J /(*-«) gu)du . 0 0
For some cases of isomorphic representations of the multiplier space (Ι,Ζ)Λ see [4], [9] . Here we formulate some of them:
(a) Let xQ(f) = aQf(0)+b0f'(0), x(f) = Ff) + G(f) with F, G € C* where 1
Hf) = j /^Ψ> φ^β 7 norm has at least one point of discontinuity a;Q€(0,l] if bQ Φ 0 and x € [0,1] if bQ = 0. Then:
BVl *j (L1 , L 1)* if ^o^O and BV\ =χ (L1 , L1 ) Λ if £Q = 0 d f V V
(57j = /GBP1 :f(0) = 0). (b) Let χ0(/)=α0/(0)+Ζ?0//(0) with b0 Φ 0,
1 x(/)=/(*o)+ [ ^
0 with h€BV. Then £7ηοΓιη =j- (L1,L1)^. (The case &Q = 0 is open problem
till now. The author managed to prove only the isomorphic embedding BV cT (L1,^1),).
norm J v ' '*'' v 1 (c) Let X0(f)=a0/(O)+^0//(O), X(/) = \ fh with /z€B7. Then L1 =r (Ll,Ll) , C*=j ( C C ) * . v . v The classical Sturm-Liouville problem belongs to above cases (a) and (b) if at least one of the boundary value conditions contains derivative, i.e. if bQ Φ 0 or b,^0. According to our remark in (b) the case of boundary value conditions Xn(/)=/(0) , x(/)=/U) is partially solved till now.
REFERENCES
1. L. Berg, Generalized convolutions, Math. Nachr. 72 (1976), 239-245. 2. N. Bozhinov and I. Dimovski, Boundary value operational calculi for
linear differential operators of second order, C.R. Aead. Butg. Sei. 31 (1978), 315-818.
3. N. Bozhinov, Differentiation properties and multipliers of Berg-Dimovski convolution for the differentiation operator, SERDICA Bulg. Math. Publ. 6. (1980), 219-239.
4. N. Bozhinov, Differentiation properties, multipliers and commutants related to Dimovski's convolutions for second order differential operators, C.R. Aead. Bulg. Sei. 34 (1981), 1057-1060.
5. N. Bozhinov, Classes of multipliers of Berg-Dimovski convolution and real Dirichlet expansions, C.R. Acad. Bulg. Sei. 34 (1981), 1213-1216.
6. N. Bozhinov, Representation formulas for the multiplier sequences of real Dirichlet expansions, C.R. Aead. Bulg. Sei. 35 (1982), 877-880.
118 N. S. Bozhinov 7. N. Bozhinov, A representation of the multipliers of Sturm-Liouville
expansions, C.R. Acad. Bulg. Sei. 35 (1932), 739-742. 8. N. Bozhinov, A convolutional approach to the multiplier problem connec
ted with generalized eigenvector expansions of an unbounded operator, SERDICA Bulg. Math. Publ. 8 (1982), 425-441.
9. N. Bozhinov, On a class of (L1 »L1) Sturm-Liouville multipliers, commu-tants and convolutions, C.R. Acad. Bulg. Sei. 36 (1983), 867-870.
10. N. Bozhinov, On a class of (L1 ,L1)-multipliers, commutants and convolutions connected with the differentiation operator and real Dirichlet expansions, C.R. Acad. Bulg. Sei. 36 (1983), 739-742.
11. N. Bozhinov, Isomorphical representations of multipliers of the smooth Berg-Dimovski convolution, Mathematics and Educations in Mathematics, Sofia 1984.
12. N. Bozhinov, On some approximation properties and multipliers of Berg-Dimovski convolution for the differentiation, C.R. Acad. Bulg. Sei. 37 (1984), 237-290.
13. N. Bozhinov, Cauchy convolutions and the multiplier problem of root vector expansions, Constructive Theory of Functions, Sofia, 1984.
14. N. Bozhinov, Operators with convolutional multiplier resolvent, Complex Analysis, Sofia, 1985.
15. I. Dimovski, Convolutions for the right inverse linear operators of the linear differential operator of the first order, SERDICA Bulg. Math. Publ. 2 (1976), 32-86.
16. A. Leont'ev, On the properties of sequences of Dirichlet polynomials convergent on a segment of the imaginary axis, Izv. Akad. SSSR ser. mat. 29 (1965), 269-328 (Russian).
17. D. Przeworska-Rolewicz and S. Rolewicz, The only continuous Volterra right inverses in Cc[0 , 1] of the operator d/dt are J , Colloquium Math. (to appear). a
18. L. Schwartz, Theorie generale des foneutions moyenne periodiques, Ann . of Math. 48 (1947), 357-929.
19. L. Schwartz, Etude des sommes d'exponentielles, Paris, 1959. 20. A. Sedlezkij , Biorthogonal expansions of functions in series of expo
nents on intervals of real axis, Uspehi Math. Nauk 37 (1982), 51-95. (Russian).
COMPARISON METHODS FOR EIGENVALUES OF SEMILINEAR ELLIPTIC
OPERATORS
R. CHIAPPINELLI Dipartimento di Matematica, Universita Delia Calabria,
87036 Rende (CS), Italy
Let Ω be a bounded open s e t i n wP (p > 1) wi th smooth boundary 3Ω.
I t i s we l l known t h a t the l i n e a r problem:
-hu = \u in Ω (1)
u = 0 on 3Ω has an infinite sequence of positve eigenvalues λ (n= 1 ,2, . . . ) , with λ -*<» as η-χ», to which, due to linearity, correspond eigenf unctions u
2 of any prescribed L norm. A classical problem in this topic is to find the asymptotic distribu
tion of the eigenvalues λ . After the pioneering work of H. Weyl [7], the systematic use of the minimax principle allowed Courant (e.g. [3]) to prove that, as n - + <* ,
λ0 = Kn2/p + 0(n1/piogn) , (2)
where κ= (2π)" (ω |Ω|)_ 1 , ω denoting the volume of the unit ball in TSr
and |Ω| the volume of Ω, and that the same result holds true when perturbing the Laplacian with a term of the form qx)u , with q£L (Ω) .
Let now /: ΩχΙΚ-^IR be a continuous function with f(x,Q) = 0 for all x G Ω and consider the perturbed semilinear eigenvalue problem
-hu + f(x,u) = k in Ω (3)
u = 0 on 3Ω .
A convenient generalization of the above classical results is given by the following:
THEOREM 1 . Suppose \f(x9u)\ < a\u\ + b and fx,-u) = - f(x ,u) for all (x, u) £ Ω Χ R and some constants a > 0 , b > 0 . Then given any r > 0 , (3)
119
120 R. Chiappinelli possesses infinitely many eigenfunctions u (r) (ft = 1,2,...) with
j u2(r) = r n Ω whose corresponding eigenvalues satisfy λ (r)-> + oo ££ n -»<», and indeed
λ (r) = κ^22/ρ + 0(ftl/p log ft), (4)
where κ is as in (2), The existence part follows from an adaptation of well-known Liusternik-
-Schnirelmann techniques. Namely, we consider the "energy functional" associated to (3) (integrals are over Ω, unless otherwise stated):
φ(Μ) = \ | |Vu|2 + | F(x,u) , (5) where ^
F(x9t) = [ f(x9s) ds , '° 0
and consider weak solutions of (3), i.e. elements of uEW1»2^) such that φ' (u)v = [ VwVt> + I f(x9u)v = λ J uv
0 for all ^€ί^1>2(Ω), where φ7 (w) denotes the Gateaux derivative of φ at the point u.
Therefore weak solutions of (3) with L2 norm equal to r>0 are critical points of φ subject to the constraint / u2 = r 2, i.e. points
u£Mr = luew1*2 (Ω) : g(u) := \\u2 =\\*
where φ' (u) = Xgf(u) for some Lagrange multiplier λ; the corresponding value of the energy functional is called a critical level of φ .
The existence of infinitely many critical points for φ relative to M
can be proved with standard methods from Liusternik-Schnirelmann theory; we refer for this, as well as for the terminology and notations, to the survey papers [4],[5] by Rabinowitz and bound ourselves to state the following result which is proved in [1],
LEMMA 1 . Under the assumptions of Theorem 1, the functional φ is bounded below on M (r > 0) and satisfies the Palais-Smale condition on M .
r .—_ v
For n - 1,2,..., let K (r) denote the family of all compact, symmetric subsets of M whose genus is equal to n. Then the numbers
c O ) = inf sup 2φ(ι/) (n = 1 , 2 , . . . ) (6) n KJr) K
n (where K runs through K (r)) are critical levels of 2φ on M ; namely, there exist u (r) € M and λ (r) £ TR such that
n r n
Eigenvalues of Semilinear Elliptic Operators 121
φ'(«„<?)) = XnW g'(unM) (7a)
24>(un(r)) = CM(P) . (7b)
The proof of Theorem 1 is now reduced to comparing the eigenvalues λ (Ρ) with the eigenvalues λ of the linear problem (1). To do this, the first step is the following:
LEMMA 2. Let λ_ be as above; then, for all v> 0 and all n = 1,2,...
r2 λ° = inf sup 2 φ (w) , (8) n K O) K °
n where
, («>=! | | v u | and K (r) is as in Lemma 1 .
77 PROOF, This follows from the properties of the genus [4] and the Courant minimax principle. See [1]. LEMMA 3. Under the assumptions of Theorem 1, there exist constants C , C > 0 (depending only on / and Ω) such that
n n
for all r > 0 and all integers n.
< C1 + C2 r -1 (9)
PROOF. First observe from (6) and (8) that
o (r)-r2\° n n < 2 sup M„
Χ Μ ) - Φ 0 ( Κ ) < 2 sup F (x , u)
νδ and therefore σ ( P ) - P 2 A ° w n < ar + 2&r (10)
because if \ f(x 9u) \ < a\u\ + b , then |F(a;,w) | <j \u\2 + 2?|M| and then the last inequality follows from Schwarz inequality (with b =b\ü\1'2) and the definition of Mv .
Next from (7a) we get in particular
4>'(un(r)) un(r) = Xn(r) £'("„(*)) «n(r)
= λ (r) r2 n
so that, using also (7b), it follows
ön(r)-r2Xn(r) = 2φ(^(ρ)) - Φ 7 ( ^ ( Γ ) ) u^v)
= J V^(r) + 2 j F(x9un(r))
- | | v u n ( r ) | 2 - j /(tf,Mn(20)wn(r)
122
and t h e r e f o r e
c (p) — v X (r) n n
R. Chiapp ine l l i
ς sup 2 F(x,u) —\f(x,u)u ( 1 0
< lav + 3bv
by easy estimates like above. On using together (10) and (11) we get
v2 X (r)-r2X° n n
< 3av + bbv (12)
which proves the claim, with C = 3a and C2 = 5b . To finish the proof of Theorem 1, just notice that the above estimate
implies λ (r) = λ° + 0 (1) as n->oo n n
for any fixed v > 0, so that (4) is an immediate consequence of the corresponding formula (2) for λ .
n One further comparison argument, which only exploits monotonicity techniques, can be given as follows: PROPOSITION . Assume there exist ql , q2 € Ζ,°°(Ω) , with q Ax) > C> 0 for some constant C > 0 and a. a. x € Ω, such that
qY(x)t < f(x,t) < q2(x)t 1 2 for a l l t>0 and a . a . χ € Ω . Let λ , λ denote r e s p e c t i v e l y the e i g e n -n» n 1: 1 , s—
values of the linear problems:
— ku + q . (x) u - Xu in Ω , u = 0 on_ 3Ω (i = 1 , 2) .
Then the eigenvalues λ (r) of (3) satisfy
d'1 X1 < X (r) <dX2 n n n
for a l l n £ l and r > 0 , w i th <£ = II ^ II C~l .
PROOF. From ( 1 3 ) , on i n t e g r a t i n g and m u l t i p l y i n g by t one h a s :
q (x) t2 < 2F(x,t)< ς (a;) t2
qx(x) t2 <f(x9t)t< q2(x) t2
(13)
(14)
(15)
(16)
(17)
for a.a. x € Ω and all t € TR (notice that F(x,t) and f(x,t)t are even functions of t) . Therefore, since q lq < d by the definition of d, we also have
d~lf (xyt)t < 2F(ar, t) <df(x9t)t
as in (5) and with the linear problems (14), i.e
(18) Now let φ be as in (5) and let Φ1,Φ2 t>e the functionals associated
Eigenvalues of Semilinear Elliptic Operators 123
(i = 1 ,2), Then (16) implies that
φ (Μ) < φΜ < φ (w)
(19)
for all u whence, on taking inf sup of each term (and considering K (r) K
ΎΙ that Lemma 2 obviously extends to characterise the eigenvalues of (14)),
r2 X1 < a (r) < r2 X2 . n n n
On the other hand, (18) implies that
d~l §'(u) u < 2cj>(w) < d$'(u) u for all u.
This inequality, with u (r) replacing the generic u, gives (on using (7a), (7b))
d l v2 X (>) < c (r) < dv2 X (r) n n ^ n (20)
The result now follows on combining (19) and (20).
Remark 1. The results described above hold essentially unchanged when the Laplacian is replaced by a general second order formally selfadjoint elliptic operator L :
/ ^ — \ a . Ax) T:— -L· dx. V ^»J dx J i,3 v 3
with bounded measurable coefficients an and a . . 0 ^,ΰ
Remark 2. If we assume that \ f (x ,u) \ < a \u\ (i.e. b = 0 in the growth assumption of Theorem 1) then the bound in the estimate (9) does not depend upon v : we have indeed
a. . (t,j = 1,2,...p) υ id
X (r)-A n n
< C, 3a . (21)
Remark 3. The case p= 1 shows some peculiar features. Without loss of generality, we can take Ω= ] 0 , π [ so that we are dealing with the semilinear Sturm-Liouville problem:
\-u" +f(xyu) = Xu , 0<χ<τ\ (22)
u(0) = κ(π) 0 . 0 2 Observe that in this case λ = n (n = 1 ,2, n .) .
Assume further that \f(x,u) \ <a\u\ as in the previous Remark. Then, using the spectral properties of selfadjoint operators, it is easy to see that (22) has no nontrivial solution when λ t in2 -a , n2 + a] (n=l,2,...): notice that these intervals are disjoint for n large enough [2]. Together with (21) this shows that actually, at least for large ft,
124 R. Chiappinelli
λ ( ζ θ - λ η η
< a .
What is more interesting is that the same result can be proved without any oddness assumption on /. To see this, one has to use rather different methods, namely degree theory rather than variational methods, and to exploit the fact that the eigenvalues of Sturm-Liouville problems are simple, i.e. the associated eigenspace has dimension one. We refer the interested reader to [2]. Remark 4. Recall that λ is said to be a bifurcation point for (1) if given any ε>0 there exists an eigenvalue — eigenfunction pair (λ tu) of (1) with |λ — λ0|<ε and 0<llwll <ε, where the norm II II is taken in some suitable function space.
It is a classical result that, if f(x,u)/u-*0 as u-> 0 , then every eigenvalue λ is a bifurcation point of (1) (see e.g. Rabinowitz [4], [5]).
Our results show that, if the latter assumption is replaced by | f(x ,u) | < a | u | , then given any r> 0 there exist (λ , u) with | λ — λ | < 3α and / u = r2> 0 solving (1): in other words, there is bifurcation (in L2) from intervals around the eigenvalues λ . See [6] for related results on
n this topic.
REFERENCES
R. Chiappinelli, On the eigenvalues and the spectrum for a class of semi-linear elliptic operators, Boll. Un. Mat. Ital. (6) 4-B (1985), 867-882. R. Chiappinelli, On eigenvalues and bifurcation for nonlinear Sturm--Liouville operators, Boll. Un. Mat. Ital. (6) 4-A (1985), 77-83. R. Courant and D. Hubert, Methods of Mathematical. Physics, Vol. I, Interscience, New York, 1953. P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mt. J. Math. 3 (1973), 167-202. P.H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, in: Eigenvalues of Nonlinear Problems, CIME (Cremonese, Roma, 1974), 141-195. K. Schmitt and H.L. Smith, On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), 294-319. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1911), 441-469.
REGULARITE DES SOLUTIONS DE L'EQUATION DES MILIEUX POREUX EN UNE
DIMENSION D'ESPACE
M. S. MOULAY et M. A. MOUSSAOUI Department of Mathematics, Houari-Boumedian University,
Algiers, Algeria
I n t r o d u c t ion
Le but de ce travail est d'etudier les proprietes de regularite des solutions de l'equation des milieux poreux en une dimension d'espace. II s'agit du probleme de Cauchy:
(li= τ^ (*(u)) ( x ' ° e R x ]0'T[
I u(x,0) = u Q ( X )
ou φ est une fonction possedant un certain nombre de proprietes qui seront precisees par la suite, et un une donnee peu reguliere.
> 1 L'existence pour une donnee u dans L (R) d'une solution au sens des distributions est prouvee par exemple dans Brezis [6], utilisant la theorie des semi-groupes non lineaires.
Le cas ou u. est plus reguliere, est traite dans B.F. Knerr, [12], qui prouve l'existence, l'unicite et une certaine regularite, pour les solutions. On peut, grace aux resultats de Kalashnokov [10] montrer que la regularite prouvee dans Knerr [12] est optimale, dans un sens que nous precisons egalement plus loin.
Ce que nous voulons prouver ici est que les solutions faibles obtenues avec des donnees peu regulieres, possedent cette
125
126 M. S. Moulay et M. A. Moussaoui
regularite optimale dans R x ]τ,Τ[ , τ > 0.
Cette recherche, menee dans le cadre du projet 0NRS: 78 AA 02 02. a deja donne lieu ä un premier resultat publie en 1984 par S. Benachour et M.S. Moulay [5] ou l'on a traite le cas ou (f)(u) = u , m > 1. Ceci est done une generalisation de ce resultat et de celui de Aronson [1].
I. - Position du probleme et notation
Soit T > 0. On notera S = R x ]0,T[
Pour u donee dans L (R) Π L ( R ) , un ^ 0 p.p., on considerera le probleme de Cauchy:
(C)
'u = φ(u) dans S t Ύ xx
u(x,0) = un(x) dans R
oü φ est une fonction numerique definie sur R , reguliere et verifiant les proprietes suivantes:
1.1 φ(0) = φ»(0) = 0 1.2 φ ( 8 ) > Ο , φ ' ΐ ε ) > 0 , φ ' ^ ε ) > 0 p o u r s > 0
1.3 f] i ^ l d s < -(H) Jo s
A / S ( j ) " ( s ) 3 s . . * n + 1.4 s -> , . ,—v— e s t d e e r o l s s a n t e , b o r n e e s u r R φ (s) s2 φ ' (s) 1.5 s -> ——,—v est bornee sur un voisinage de 1T φ( s) . . &
origine
Remarque 1.1.
La condition (1.4) peut etre affaiblie, en ce sens qu'il suffit qu'elle soit verifie sur ]0, Munll^ [> e t pour la regularite de φ on peut prendre:
Equation des Milieux Poreux en une Dimension d'Espace 127
[> e C 3 ( R + ) n c 1 ( R + ) .
Nous voyons que (j)(u) = u , m > 1, verifie les conditions ci-dessus.
~ T r ^ . m m-1 ^. . . . Dans ce cas la fonction v = —^-r u joue un role particulier. La fonction v represente en effet la vitesse de filtration des x r
particules du liquide ou du gaz dans le milieu poreux, u designant la saturation. Aussi nour introduisons l'analogue dans notre cas et qui est alors definie par:
(*) v = ψ(υ) et ψ(8) =/ )τ(ξ) άξ, s > 0.
Le probleme (Q) est alors formellement equivalent a:
(C) v = σ(ν) v + v2 t XX X
v(x,0) = vQ(x)
avec vn = ip(un) et a(s) = _jrHs)_ (ψ-1)1(s) pour s > 0.
Dans la suite u et v seront toujours reliees entre elles par (*).
La methode que nous utiliserons reposera sur les proprietes des solutions telles qu'etudiees par Knerr [12] et sur des estimations ä priori que nous etablirons sur ces solutions.
Donnons tout de suite la definition d'une solution faible du prob 1 erne (Q) :
Definition 1.1.
Soit u E L ( R) Π L ( R ) , u ^ 0.
On dira que u est une solution faible de (Q) si: a) u G L2(S) ; u ^ 0.
b) φ(ιι) €= L2(S)
128 M. S. Moulay et M. A. Moussaoui
c) if (uft - Φ(^)χίχ) d x d t = -/"f(x,0) uQ(x)dx S R
1 pour tout f dans H (S) tel que f(x5T) = 0.
L'existence d'une solution de (C) peut etre prouvee de deux manieres differentes. La premiere consiste ä approcher un pa une suite de fonctions regulieres strictement positives sur R Cette methode permet d'etablir une inegalite analogue ä celle prouvee par Ar onson-Beni Ian [3] dans le cas oü <£(u) = u , et est donnee par la:
Propos it ion 1.3.
Soit u une solution faible de (C)· 0 n pose
1 0 " 2 + inf i£<^
+ Φ1(s) R
On a : u, > - a ^ + ^ - ^ - u2 . t t u x
Remarque:
On retrouve dans le cas ou (f)(u) = u , α = r , comme dans [3 m+ 1
PREUVE:
La preuve suit la meme idee que dans [3].
Soit v = ijj(u) comme definie precedemment pour une solution de (C) assez reguliere.
Posons p = v . La fonction p est alors solution de: xx L(p) = pt - [φ'<η)ρχχ + 2(φ·(υ) + ν ) χ Ρ χ + (2 + ^^±
φ (u) u x r
= 0.
Equation des Milieux Poreux en une Dimension d'Espace 129
G r a c e ä l ' h y p o t h e s e (H) ( 1 . 4 ) on a :
L( " f ) * 0
et par le principe du maximum :
a t
On obtient alors lfinegalite annoncee en remarquant que v est solution de :
v = φ'(u)v + v2
t XX X
La deuxieme methode pour prouver l'existence d'une solution faible pour (C) consiste a approcher u par des fonctions regulieres dont le support est un Intervalle et ä utiliser les resultats de Knerr. Nous adoptons cette methode afin dfetablir les estimations ä priori necessaires pour un passage ä la limite.
2 . - Solutions Regulieres et Estimations a priori
L'unicite de la solution faible peut etre prouvee exactement de la meme maniere que dans [14].
On considere ä present une suite (u ) ><| de fonctions regulieres tell es que :
u ^ 0 E sur R on
u (x) > 0< >x ] - n, n [ on
u converge vers u_ dans L ( R) , 1 ^ p < on U
l S \ u n = M.
Alors d'apres [12], pour tout n £ 1, le probleme
dans S
" \u(x,0) = u (x) ( ( „. dans R
130 M. S. Moulay et M. A. Moussaoui
admet une unique solution faible u ayant les proprietes suivantes:
(P J : 0 ύ u < M , u G C°("S) . 1 n n
(P ) : II existe deux fonctions λ^ et λ^ dans C ([0,T)
Π C ' ([τ,Τ]) pour τ > 0, verifiant :
(i) : X,(0) = - n , λ1 decrois sante.
(ii) : λ (0) = n , X croissante.
(iii): u (x,t) > 0< >(x,t)e[(x,t)
RxR+;A^(t)<x<X2(t)
= P ([unD,
(iv) : u est une solution classique de (C ) n n dans P([u ]).
(P„) : la fonction v associee a u verifie : J n n
(i) : |vn(x,t) - v n(x',t)| S C |x-x»|
pour x,xf dans R et t dans [τ,Τ]; C etant une constante dependante de τ, Μ = sup|u (x)| et
xER N = ΙΙΦ^οη^Ν.ί τ > 0 ;
(ii) : si v est 1ipsehitzienne de constante on K sur R, alors la propriete (i) est verifiee dans R x [0,T] avec C dependante de M, K et N.
On appellera P, les proprietes ci-dessus. Dans la suite on omettra lfindice n et on posera :
- u au lieu de u , v au lieu de v . n n
λ. (t) au lieu de λ. (t) , i = 1 ,2
Equation des Milieux Poreux en une Dimension d'Espace 131
- Ω = S τ < t < T pour 0 < τ < T.
" ( P i u ] ) e = ( x , t ) ; X ^ t ) + ε < x < X 2 ( t ) - ε ; 0 < t < T
♦ t
( P [ u ] ) c
x = X , ( t )
/ /
x = X 2 ( t )
-n - n + € n -6 n
On utilisera aussi une function de troncature Θ reguliere et teile que :
0 < Θ < 1 , Θ G & (P(u])>
Θ Ξ 1 dans P([u])
ΘΨ= 0 au voisinage des courbes X..(t) et X_(t)
l©x I ί £ , |0J £ § ; |Θ | < 4 x ε t e xx' ε
On notera V£ = P([u])/P([u])£.
LEMME 2.1.
(i) / u(x,T)dx = / un(x)dx : (conservation de la masse) J R J R U
( Ü ) i y u 2 (x ,Ddx + ff φ·(υ) u2 dxdt = |y*u2( x )dx.
132 M. S. Moulay et M. A. Moussaoui
PREUVE:
( i ) On a : ff u Θ d x d t = ff φ ( ι ι ) 0 d x d t . ;;0 Ü JJ tt x x
τ τ u Θ etant reguliere, on a en integrant par parties en t :
ff ut0 dxdt =f u(x,T)0(x,T)dx - j u(x,T)0(x,T)dx - ff u0tdxdt S2 R R Ω
τ τ
de p l u s :
I ff u0 d x d t I < s u p ( u ) ff | 0 I d x d t . JJ v V V
ε ε ε
Le dernier membre de lfinegalite ci-dessus tend vers 0 lorsque ε tend vers 0.
Par ailleurs
f u(x,T)R(x,T)dx et f u(x,t)R(x,t)dx R ^R
t e n d e n t r e s p e c t i v e m e n t v e r s
/ u ( x , T ) d x e t / u ( x , T ) d x quand ε + 0 JR JR
Egalement :
\ff φ ( u ) χ χ 0 d x d t \ = \ff φ ( u ) χ 0 χ d x d t I Ω τ
JfQ < Hr- ux ) u e x d x d t I £ \ffQ vxuexdxdt τ "JT
< Κ(τ) sup u ff |0 |dxdt V "" V ε ε
le dernier terme ci-dessus tend vers 0 lorsque ε tend vers 0, Par suite on a :
J u(x,T)dx = f u(x,i)dx
Equation des Milieux Poreux en une Dimension d'Espace 133
M a i s p a r u n i f o r m e c o n t i n u i t e de u s u r t o u t c o m p a c t de S, on a :
l i m / u ( x , x ) d x = / u n ( x ) d x ; d ' o u ( i ) τ+O • ' R ^ R υ
(ii) En multipliant l'equation des milieux poreux par u0 et en utilisant les memes arguments que precedemment on obtient la deuxieme egalite.
ru
Soit <£(u) = / φ(&)ά&, la primitive de la function φ s'annulant J 0
en 0. Grace ä la croissance de φ on a :
Φ(ιι) ύ ιιφ(ιι)
et le :
LEMME 2.2.
f Φ(ιι(χ,Τ)) + ff (φ(ιι))£ dxdt =f Φ(u0(x))dx <f u0φ(u0)dx
PREUVE:
En multipliant l'equation u = <\)(u) par Θ φ(υ), il vient en integrant sur Ω :
ff u φ(ιι) Θ dxdt = ff φ(ιι) φ(ιι) Θ dxdt JJ ü t JJ ü X X
τ τ Notons I1 et I? les deux membres de lfegalite ci-dessus. On a :
I , = ff - ^ ^ ( u ) ) 0 d x d t = f Θ Φ ( ι ι ( χ , Τ ) ) ά χ - / θ Φ ( ι ι ( χ , τ ) ) < Ι χ - Φ(ιι)Θ<1χ<1ϋ 1 JJÜ d t ; R JR Ω Ü
τ τ
\ff Φ(ι ι)Θ d x d t | ύ ff ι ιφ( ι ι ) |Θ I d x d t ύ s u p ιαφ(ιι) ff |Θ I d x d t JJÜ t JJ t JJV z τ ε ε ε
Cette derniere quantite tend vers 0 avec ε. DToü :
134 M. S. Moulay et M. A. Moussaou i
l i m I = / " φ ( ι ι ( χ , Τ ) ) α χ - / * Φ ( ι ι , ( χ , τ ) ά χ ε->0 R
D ' a u t r e p a r t :
Xo = - IT (Φ(") )20dxdt - ΛΓ φ(ιι) φ(ιι)Θ dxdt ^ " Ω χ Ω X X
τ τ Le deuxieme terme du membre de droite de lfegalite ci-dessus peut etre reecrit :
ff (Φ' ( u ) u ) υφ(υ) 0 dxdt JJ n u x x
et tend vers 0 avec ε comme on lfa vu precedemment. Par consequent :
lim I . = - ff φ(υ)2 dxdt
A i n s i on a
f Φ ( ι ι ( χ , Τ ) ) α χ + ff (<f>(u) ) 2 d x d t = f Φ ( ι ι ( χ , τ ) ) ά χ J R JJ SI X ^ R
τ et on obtient en faisant tendre τ vers 0 l'egalite :
f Φ(ιι(χ,Τ))<Ιχ + ff (φ(η)χ)*άχ = f Φ(υ0(χ)Ηχ
LEMME 2.3.
(i) u (.,T) G L ( R) pour tout τ > 0, avec
/ |ut(x,T)|dx £ ψ ||u0|| , = C T
(ii) φ(υ) est 1ipschitzienne sur S = S t ^ τ pour tout τ > 0 et on a :
)(u(x,x)) - φ(υ(χ',τ))| < C |x-x'
Equation des Milieux Poreux en une Dimension d'Espace
PREUVE
Remarquons d'abord que pour t > 0 fixe, (φ(ιι)) est continue x. II suffit de voir que φ(υ) = v u, et dTutiliser les prop etes P ,P . Par suite :
r / 2 ( t )
/ u (x,t)dx =/ φ(ιι) dx = φ(υ) (x,t) JR Ü J \<\(t) X X X
X2(t)
X^t)
d'oü en posant u (χ,τ) = sup (u (χ,τ), 0) et u = u - u il
f u+(χ,τ) = f u (χ,τ)άχ
Compte tenu de la Proposition 1.3 on a
u (χ,τ) ύ — U ( X , T ) , d'ou :
f |ut(x,T) |dx = f u*(x,T)dx + f u~(x,l)dx = 2fu~(x,T)dx <~\ |uc
d'oii (i).
Pour (ii) soient x G R , τ > 0 . On a :
φ(υ) (χη,τ) = / ° φ(ιι) (ξ,τ)αξ = / ° u (ξ,τ)αξ
et d'apres (i) on a :
λ^τ) Xt(T)
>(u)x(x0,T)| £ ψ ||u0l = C
d'oii
) ( U ( X , T ) ) - φ Ο ( χ ' , τ ) ) | ύ C T |x-x'|; V χ,χ' R.
LEMME 2.4.
On a :
136 M. S. Moulay et M. A. Moussaoui
ff t(f)? (u)u2. dxdt + I f 4>(u)£ (x,T)dx ύ f u0φ(u0)dx
PREUVE
Multiplions l'equation des milieux poreux par t(<j>(u) )Θ et integrons sur Ω ; on obtient :
X1 = ff t^'iu^l Odxdt = ff ίφ(υ)χχφ(η)ι. Gdxdt = I2 τ τ
Integrons par parties en x dans I „
I = - ff t<t>(u) φ ( ι ι ) . Θ d x d t - ff ϋφ ( ι ι ) φ ( ι ι ) φ ( ι ι ) Θ d x d t ί. JJ p. X t X JJ /-s X X t X
τ τ
" " J 1 - J 2
On a
J1 - \ ffn t 0 Ä * ( u > x 2 dxdt
τ = \ [ φ ( η ) 2 ( x , T ) 0 ( x , T ) d x - ^ / ^ ( u ) 2 ( x , < r ) 0 ( x , T ) d x
2JR X 2 i / R X
= \ff φ ( u ) 2 Θ d x d t - -ijfjT t φ ( u ) χ 0 t d x d t Ω "V τ ε
En faisant tendre ε vers 0, puis τ vers 0, on obtient :
lim ( lim J ) = | / φ < " ) χ (x,T)dx - \ff§(u)^ dxdt τ->0 ε-*0
D'autre part on a :
J2 = ff ίφ(υ)χφ(υ)ίΘχαχάϋ =ff |tutvx0x| |ιιφ»(ιι)| dxdt ε ε
Or on a : „φ.(„) . 4f(iLi Γ ϋ ϋ ΐ ] . "2Φ;(")|/-" ΦΙ^ Ι de|
ψ Φ(ιΟ L u J Φ ( υ ) r o u ' < 11 2 φ ' ( ΐ ΐ ) fU φ ' ( 8 ) d s = v UV(U)
φ θ ) J n s φ ( u )
Equation des Milieux Poreux en une Dimension d'Espace
il vient alors que : 137
|JJ ^ C sup V ε
ιι2φ' (u) <f>(u) \ ff t I u I dxdt
I J v ε
Grace ä l'hypothese H et au lemme 2.3., J?(e) tend vers 0 lorsque ε tend vers 0.
On en deduit que :
ff t <J)f(u)uj; dxdt + | / φ ( " ) * (x,T)dx = yj//^(u)* dxdt S R
et on conclut grace au lemme 2.2
3 . - Existence, Unicite et Regularite de la Solution de (C)
Dans cette partie, on repose u au lieu de u et u au lieu de n on .
u . On considere comme dans le debut de ce papier u G L H L (R)
Grace au §(II) on a le :
THEOREME 3.1
II existe des constants C1 et C? ne dependant que de M> I I uo ' Ί 1 (R) telle ue :
(i) \JIL2(S) * C1
(ϋ) Ι^Φ(πη)||Ηι(3) s c2<
Comme consequence de ce theoreme on a
THEOREME 3.2. : Soit uQ G L Π L°°(R)
Le probleme (C) admet une unique solution selon la definition 1.1.
138 M. S. Moulay et M. A. Moussaoui
PREUVE;
Grace au point (i) du Theoreme 3.1., il existe une sous-suite appelee encore (u ) qui converge faiblement vers une fonction
2 n
u dans L (S), done aussi dans D'(S).
D'autre part d'apres ( ü ) du meme theoreme, la suite (φ(υ )) 2 n
converge fortement vers une fonction w dans Lr (S). Par suite 0 loc (φ(υ )) converge presque partout vers une fonction w, et done
n -1 (u ) converge presque partout vers φ (w). n 1 °°
Or la suite (u ) est bornee dans L Π L (S); eile converge n 1 -1 fortement dans L.. (S) vers φ (w) et done aussi dans D'(S). loc
Finalement, on a :
u = φ~1 (w) dans D' (S)
2 D ^ u : ($(u )) converge faiblement vers φ(υ) dans L (S). Le passage a la limite sur n est alors loisible dans 1'equation integrale suivante :
f f u f - (<$>(u ) ) f d x d t = / u ( x ) f ( 0 , x ) d x JJ n t Y n x x • ' o n b R
pour tout f dans H (S) teile que f(x,T) Ξ 0
Par suite u est solution de (C) au sens de 1.2.
L'unicite de la solution se montre comme dans [14].
THEOREME 3.3.
La solution u de (C) possede les proprietes suivantes :
(i) t φ(υ) est dans L (S) et on a :
)(u(x,t)) - <|>(u(y,t))| S Ct |x-y|, t > 0.
Equation des Milieux Poreux en une Dimension d'Espace 139
( i i ) v e s t bo rnee dans S pour t o u t τ > 0 . x T
( i i i ) s ' i l e x i s t e une c o n s t a n t e C > Ü e t m > 1 t e l s que
1'on a i t :
φ' ( s ) > C sm~1
1 m-1
alors u G C (S ) pour tout τ > Ü. τ
(iv) u (t,.) est dans L (R) pour tout t > 0 et on a :
II·,.«:..)!! ! ^ C t L](R) Z
PREUVE:
Pour (i), on a grace au Lemme 2.3. (ii) : ^(un(x,t)) - <|>(un(y,t))| < Ct |x-y | , n > 0
mais (u ) converge p.p. vers u. On en deduit (i) par passage a la 1imite sur n.
Pour (ii) soit τ π > 0 fixe. Posons :
v = ψ (u ) n r n , v = ψ(ιι)
Grace a lfunicite des solutions, la restriction de u ä S est n το
la solution du probleme :
<c T ) το
Wt = * ( w ) x x dans S
w(x,TQ) = un(x,TQ)
La donnee initiale de (C ) est continue, bornee et teile qui
φ(υ (.,τπ)) soit 1ipsehitzienne de constante
C = C(xQ, a, ||uQ|I . ) independante de n.
On a done d'apres [12] :
140 M. S. Moulay et M. A. Moussaoui
| v n ( x , t ) - v n ( y , t ) | < C T | x - y | , t > TQ ( x , y ) R2
On obtient alors (ii) par passage a. la limite sur n dans integrale ci-dessus.
Pour (iii) il suffit de remarquer que :
m-1
Pour (iv) la suite (u )_(.,t) . Λ est bornee dans L ( IR) pour n t n^ U tout t 0 < t ύ Τ, grace au lemme 2.3.
Elle admet par consequent une sous-suite qui converge vers une mesure de Radon u(t) pour t > 0 et u(t) = u , au sens des dis tr ibutions.
Elle verifie en plus l'inegalite
(*) s IKII * c ||u || , t > o t JHWi) L ] ( 1 R )
,y^( IR) e t a n t l ' e s p a c e d e s m e s u r e s de Radon s u r IR .
De i) ci-dessus, il vient que u est continue sur S. Par suite SQ = (x,t), u(x,t) * 0 ei solution classique de (C). Sn = (x,t), u(x,t) * 0 est un ouvert dans lequel u est
Dfoü : u = u x ou χ est la fonction caracteristique de Sn. t t bQ bQ u
Done u est dans L, (S) et le resultat est une consequence de t loc n
THEOREME 3.4.
Soit u_ E L (R) et u la solution de (C) au sens des distributions (c.f.: [6]).
Equation des Milieux Poreux en une Dimension d'Espace 141 Alors pour tout τ > 0, u possede les memes proprietes que la solution de (C) au sens de la definition 1.2. dans S .
PREUVE:
La solution u de (C) au sens des distributions est d'apres [6] dans C°([0,T[, L (R)) Π L°°([T,T|XR) pour tout τ > 0.
Elle coincide, par unicite* des solutions, avec celle du probl«
( C T ) w = (b(w) d a n s S
t Y xx τ
I W ( X , T ) = U ( X , T ) d a n s R x τ
au s e n s de l a d e f i n i t i o n 1 . 2 .
P a r s u i t e u p o s s e d e l e s p r o p r i e t e s e n o n c e e s d a n s l e T h e o r e m e 3 . 2 .
Remarque. Meme en supposant l a donnee i n i t i a l e u„ dans ( ) l a f o n c t i o n v 1 i n t r o d u i t e au § I e s t l i p s c h i t z i e n n e mais non C . C f e s t en ce sens que l e
r e s u l t a t du theoreme 3.3 e s t o p t i m a l .
B I B L I O G R A P H I E
1. D.G. Aronson, Regularity properties of flow through porous media, SIAMJ. App. 17, (1969), 461-467.
2. D.G. Aronson, Regularity properties of flow through porous media, a counterexample. SIAMJ. App. 19, (1970), 299-307.
3. D.G. Aronson and P. Benilan, Regularite des solutions de l'equation des milieux poruex. C.R. Aoad. So. Paris (1979) 288.
4. J. Bear, Dynamics of Flouds in Poious Media, New York, American Elsevier Publishing Company - Inc. 1972.
5. S. Benachour,and M.S. Moulay, Regularite de la solution de l'equation des "milieux poreux" en une dimension d'espace. C.R. Aoad. So. Paris (1984), 298.
M. S. Moulay et M. A. Moussaoui
H. Brezis and M.G. Crandal, Uniqueness of solutions of the initial - value problem for u - Δφ(ιι) = 0. Jour. Math -
Pures et Appl. 0021 - 7824/1979/153. R.H. Brooks and A.T. Corey, Properties of porous media affecting fluid flow. Journal of irrigation and drainage -
division ASCE 92, (1966), 61-88. B.H. Gilding, Holder continuity of solution of parabolic type. J. London Math. Soc. 12 (1976). S. Irmay, On the hydraulic conductivity of unsaturated soils Trans. Amer. Geophys. Union 35 (1954), 463-467. A.S. Kalashnikov, The occurrence of singularities in solutions of the non-steady seepage equation - Zh. Vychisl. Mat : i Mat Fiz. 7, 440-444 (1967) (translated from USSR
computational) Mat - Phys. 7, 269-275 (1967). S. Kamenomostkaya, Israel J. Mat. 14, (1973), 76-78. B.F. Knerr, The porous medium equation. Trans. Amer. Math.
Soc. 234 (1977).
THE SMOOTHNESS OF THE HEAT EQUATION SOLUTION IN A SINGULAR DOMAIN OF i? n + 1
B.-K. SAD ALL AH Ecole Normale Superieure, 16050-Kouba, Algiers, Algeria
7Ϊ+ 1
1. Introduction. Let Ω be any bounded convex polyhedral domain in R , generated by the variables (t, x) , where x = (x L, · · · , x). We denote by Γ the boundary of Ω and by Γ„ the part of Γ which satisfies:
(Τ,χ)ΕΥ , where T = sup t. (t,x)ET
For the definition and properties of convex polyhedral domains, see for example Glazman and Liubitch [1],
We are concerned with the following problem: The existence and the tion u£H * (Ω) of :
D.u-uL u- · · · - D2 u = f €£2(Ω) t x\ x J x
1 2 uniqueness of the solution u£H * (Ω) of :
(p ) i
where 51,2(Ω)= jatff'tß) , D 0χ uei2(U) ; ΐ , j = 1 , · · · , n .
2 V 3
L (Ω) being the space of the functions the squares of which are integrable in 1 2 Ω. Also, the space (of Sobolev) H (Ω) is a space which, when equipped
with the norm :
Νΐ«1'2^) = ( ι ΐ " ΐ & ( β ) + Σ KA.MII2)
becomes a Hubert space. We note that || · || (without index) denotes throughout this paper the L
norm. We remark that the boundary condition of problem (P) is justified by
the theory of symmetric systems of Lax and Phillips [6] and the theory of
143
144 B.-K. Sadallah Kohn and Niremberg [4].
Problem (P) has been solved in the case of one space variable (ft = 1) by the elliptic regularization method (Sadallah [11]) and by the approximation method of the domain Ω. (Sadallah [12]). In present paper the former approach is used because it agrees better with the method of the mathematical induction (on n) ,
Our main result is similar to the one obtained with one space variable, this is:
n+ 1 THEOREM 1. When Ω is a Pounded convex polyhedral domain of R , the heat operator D, — Δ is an isomorphism from
#^'2(Ω) = κε#1,2(Ω) , ^ ι Γ _ Γ - π1
onto L2(Q). Observe that this result is not true when Ω is not convex, even if Ω
is cylindrical with respect to the time variable. Indeed, the non-convexity of Ω allows some singular solutions to appear, hence, making the space of
1 2 the solutions larger than H ' (Ω) (see, for instance, Moussaoui and Sadallah [8],[9]). The proof of Theorem 1.1 will be achieved by induction (on n) . Since the case n—\ has been studied, see Sadallah [11], it remains to show how to pass from the dimension n to the dimension n+ 1 .
For simplicity, we restrict ourselves to exhibit the details of the passing from one space dimension to two dimensions, knowing that the general case may be treated similarly.
2. The Study of the Regularized Problem. From now on, (x,z/) will denote the two space variables. We introduce the following hypothesis on the boundary of Ω :
f Γ„ is reduced either to one point or to one segment (H)\
I (of R) This condition permits us to replace the boundary condition u\v v =0,
' N valid almost everywhere, by U\r= 0 . In the general case, (H) means that r dim Tjr,<n.
Now, consider the following (regularized) problem which is to be solved under the condition (H) :
(pe) s
Existence and uniqueness of the solution ue £Η2Ω) Π^0(Ω) of the equation:
Dtu -eD2_ue-Dxve-DyU£ = f£L2(ü) where ε > 0 .
Heat Equation Solution 145 2.1 Solution of Problem (P ).
THEOREM 2. For each positive ε and / £ j(Ω) , problem (P£) has one and only one solution.
PROOF. The domain Ω has the continuation property of Necas [10] because Ω 2 2 is convex and bounded. Hence, the operator D^: H (Ω) —> L (Ω) is compact,
consequently, the operators eD+ + D„ + D_. and D, — e D.—D^ — D^ have the same u JO y & h £C y
index. Observe that the Sobolev spaces, the convexity of domains and the condi
tion (H) are preserved under the change of variables : Ω > Ωε
(t9x9 y) . > (t//e ,x,y) . 2 2 2 2 2 2 Moreover the o p e r a t o r eD.+D +D becomes Δ = D, + D +D . The uniqueness Ό x y ~u x y
being obvious, Problem (Ρε) is equivalent to the following one:
Existence of the solution V €#2(Ωε) Π#*(ΩΓ) of the equation
(?ε) ε v ε> ον ε'
Indeed, it suffices to put :
u£(t9x9y) = ve(t/Se9x9y)
f[t 9x9y) = g t/Se,x,y) .
Finally, we know [3] that problem (P£) has a unique solution.
2.2 A priori Estimate, We shall look for an a priori estimate of the type of the second fundamental estimate of Ladyzhenskaya and Uralftseva [5], More specifically, we shall prove the following:
PROPOSITION 1. There exists a positive constant C9 which only depends on the geometry of Ω, such that :
I M f f l , 2 ( n ) + e||D*ue|| + /E [\\vtDxuz\\+ \\OtDyuc\\)<c\\f\\ ,
where u is the solution of problem (Ρε).
To derive this basic inequality, we need the results of the following 2
four lemmas: Let (·»·) be the scalar product of L (Ω) ,
LEMMA 1. One has: 2 2 2 "y"e' w"x"y ~ε i
(θ|"ε»°«"ε) = \\ΗΌχ»Λ* 2
146 B.-K. Sadallah
It is a result of Grisvard [2'J.
LEMMA 2. There exists a positive constant L which only depends on the geometry of Ω, such that :
||«e||2+ ||οχ«ε||2 + | |ζν«ε| |2<ι 11/11*.
Indeed, the boundary c o n d i t i o n u \ = 0 imp l i e s :
( / , M £ ) = ε | | ß t M J | 2 + H ^ K J I 2 + \\VyUz\\Z.
T h e r e f o r e :
Ι Ιοχ«εΙΙ2+ΙΙ^«εΙΙ2<ί| |«εΙΙ2 + · Ι Ι / Ι Ι 2 , νζ>ο. We obtain the estimate of the lemma by applying Poincare inequality
(here, u €#0(Ω) and Ω is bounded) and choosing Z small enough.
LEMMA 3. When n= 1 , there exists a positive constant M which only depends on the geometry of Ω, such that :
2e\(Dtu£,D2tuc)\ + 2\(Dtue,D2
xu£)\ < IM ||θ2 u£ ||2 + | · ||/||2, V l > 0
for all positive ε small enough. Here, x is the (unique) space variable. This lemma is a result of Sadallah [11]. The analogous of this lemma for n=2 (which is required for the induc
tion) is the following.
LEMMA 4. There exists a positive constant M' , which only depends on the geometry of Ω, such that:
2e\(Dtuz, D2tu£)\ + 2|(0tu£ , 0 ^ ) 1 + 2\Dtu£ ,D2
yu£)\
< i w ' ( | | D 2 « e | | 2+ | | D 2 u e | | 2 ) + ^ . | l f l l \ vz>o
for all positive ε small enough.
PROOF. Let us set Ω = ΩΠ y = constant, Ω = Ω Π x= constant and Γ (resp. Γ ) the boundary of Ω (resp^x).
Hence, Ω,. (resp. Ω„) is a polygonal domain in t9x) (resp. (£,£/)) and we have, for almost every y (resp. x):
u\Q eH2(tty)(\Hl(tty) (resp. u^ <Ε#2(Ωχ) n Η^ςΐχ)) .
Then, Lemma 3, yields :
2e\(Dtue,D2tue)\ +2\(Dtue,D2
xuz)\ + 2 \ (Dt u£ , D2 u£) | =
\ly2e(Vtuz,O2tuz)L2üjdy | + | / 2 ( D t u £ , D^)f Q % | +
+ l£2(D*Mc.^V(Jd,|
Heat Equation Solution 147
y · 11/11 + » ' · ι ( ΐ Ι ί » ε Ι Ι 2 + 1 Ι ^ » ε Γ
where M' = sup (M(x) +M(y)). This supremum is bounded because the functions M(x) and M(y) only depend on the positions of the faces (of Ω) with respect to the coordinates system and because the number of these faces is finite. This proves the Lemma.
Proof of Proposition 1. Expanding the expression
we get by Lemma 1
i2
\Dtu£-ED2turD2
xue-D2yu£\\2
9
= \\Dtu£\\ + ε P ; w e l l + 2e\\DtDxue\\ +
+ 2z\\DtDyue\\2 + H4« e | | 2 + Hz)Jttell2 +
+ iDtu€9Dyu£)].
Fur thermore , Lemma 4 g ive s t he e s t i m a t e
I I? - I I _ . . 2 | f l t M e l l 2 + ε 2 | | 4 Μ ε | | 2+ 2 ε | | ζ ) ί 0 χ Μ ε ! | 2
+ 2 ε | | 0 ί £ ) uf
+ 2\\DxDyuc\\2 + ^-lM>) ( | | 0 2
M e | | 2+ | | 0 2 u £ | | 2 )
< ( j - + i) l l f l l 2 , v i > o . The proof ends by choosing Z small enough (e.g. I = \ /2M/) in the previous inequality and making use of Lemma 2.
3. Proof of Theorem 1. We shall consider the two possible cases depending on whether hypothesis (H) is verified or not.
Case 1 : Ω verifies the condition (H). o For all ε > 0 and f €. L (Ω) , there exists, according to Theorem 2.
u£€H (Ω) Π# (Ω) which is a solution of the equation:
Dtu£-eDlue-Dxue-D2ue= f (1)
Since u satisfies the estimate ||wP|| ι 2 ^ ll/ll (see Proposition 1), ε b H » (Ω)
148 B.-K. Sadallah where C does not depend of ε, there exists a subsequence u . (weakly con-vergent in H »2(Ω) to a function u, when ε· tends to 0. Clearly, u£H (Ω) OHη(Ω) (because each u£y is in the same space).
2 Moreover, Proposition 1 also shows that e.D^u£. is (weakly) converts d gent to 0. Thus, putting ε = ε· in Equation (1) and making ε·->0, we get:
t/ J
Z^ w - Όχ u - Dy u = f , 1,2 1
where u € H (Ω) Π# (Ω) . This proves the existence of the solution of problem (P).
As for the uniqueness of the solution, if we assume that D,u—ku=0>
we deduce: 0 = (Dtu -tsu, u) = \\Dxu\\ + \\D w|| ,
2 2 consequently £? w = Z)7 u = 0, thus D„u = D.. u = 0 . xh Lj X Lj
The equation D^.u — Au = 0 leads to D,u = 0. The solution w is then a constant and U\„ = 0. Hence, u = 0. Case 2 : Ω does not verify the condition (H).
In this case, Γ^ is a polygonal domain. We shall reduce the problem to the previous case by a suitable extension of the domain Ω.
3 Let A be a point of R such that its first coordinate t0 verifies
Perform the projection of the point A on the plane of Γ^ parallel to the time axis. A is chosen such that its projection (on Γ*τ) belongs to Γ«.
Then, the cone C of vertex A and base Γ^ (its altitude is h = t^—T)
is convex. By choosing h small enough, it is always possible to make the polyhedron til=ttUC convex. So, Ω; is a polyhedron which satisfies the hypothesis (H).
In general case, it is suitable to express this idea in terms of convex envelopes.
2 ~ 2 Now, if / is given in L (Ω) we denote by f (€ L (Ω7)) the continua
tion of f by 0 in Ω;. Hence, according to Section 3.1, there exists 1 ? 1 ~
uEH ' (Ω') Γ\ HQ(ti') solution of D^u —Διι = f. Therefore, the restriction W|0 of u is a solution of problem (P) in Ω. This proves the existence of the solution. As regards the uniqueness, the proof is similar to the first case.
H e a t E q u a t i o n Solu t ion 149
REFERENCES
1. I. Glazman and Y. Liubitch, Analyse Lineaire dans les Espaces de Dimensions Finies, Ed. MIR, Moscow, 1974.
2. P. Grisvard, Alternative de Fredholm relative au Probleme de Dirichlet dans un polyedre, Ann, Scuola Norm. Sup. Pisa, II.3 (1975), 359-388.
3. J. Kadlec, On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of a convex open set, Czech. Math. J. 14 (89) (1964), 386-393.
4. J. Kohn and L. Niremberg, Degenerate elliptic-parabolic equation of second order, Comm. Pure Appl. Math. 20 (1974), 797-872.
5. 0. Ladyzhenskaya and N. TralTtseva, Linear and Quasilinear Elliptic Equations, Academic Press, 1968.
6. P.D. Lax and R.S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427-455.
7. J.L. Lions and E. Magenes, Problemes aux Limites non Homogenes et Applications, Dunod, Paris, 1968.
8. M.A. Moussaoui and B.K. Sadallah, Regularite des coefficients de propagation des singularites de 1'equation de la chaleur dans un domaine polygonal plan, C.R. Acad. Sei. Paris, 293 (1981), 297-300.
9. M.A. Moussaoui and B.K. Sadallah, Regularity results in the propagation of singularities for some evolution equations in a plane polygonal domain, Proceedings First International Conference Math. Gulf area (1984), 273-287%
10. J. Necas, Les Methodes Directes en Theorie des Equations Elliptiques, Masson, Paris, 1967.
11. B.K. Sadallah, Regulari£e de la solution de l'equation de la chaleur dans un domaine plan non rectangulaire, Boll. Uni. Mat. Ital. (5) 13-B (1976), 32-54.
12. B.K. Sadallah, Etude d'un probleme 2m-parabolique dans des domaines plans non rectangulaires, Boll. Uni. Mat. Ital. (6) 2-B (1983), 51-112.
SEQUENCES OF FOURIER-YOUNG COEFFICIENTS OF A FUNCTION OF
WIENER'S CLASS
R. N. SIDDIQIt Department of Mathematics, Kuwait University, Kuwait
Abstract. In this paper, we study the summability of certain sequences of Fourier-Young coefficients of a function of Wiener's class by using σ-regular matrices. Further we show that not only these sequences A-, (X) but even the modulus of these sequences \A, (a;) | and allied sequences are summable under certain hypotheses.
1. Introduction. Let σ be a mapping of a set of positive integers into itself. A continuous linear functional φ on % , the space of real boun-
Ύ oo ' r
ded sequences, is said to be an invariant mean or a σ-mean if and only if (i) φ(χ) > 0 when the sequence X = x has x > 0 for all n, (ii) φ θ ) = 1 , where e = 1 , 1 , 1 , . . . , and (iii) φ(χ , A) =φ(χ) for all x € i^.
For certain kinds of mappings σ, every invariant mean φ extends the limit functional on the space c of real convergent sequences, in the sense that φ(χ)=ϋταχ for all x € c , Consequently, c <= V . where V is the set of bounded sequences all of whose σ-means are equal. In case O is the translation mapping n->n+ 1, the σ-means are the classical Banach limits on Z
and V is the set of almost convergent sequences [3], If X = \x 1 , set TX= \x , x . It can be shown that the set 7 can be 1 n L o(n) J o
characterized as the set of all bounded sequences X for which lim (X +TX +. . . TPX)/p+ 1 P
exists in the space I and has the form LX, LX being the common value of all σ-means at X (cf. Raimi [4]). We write L = O-limX.
This research work was completed while the author was spending his sabbatical at McMaster University, Hamilton, Ontario, Canada.
151
152 R. N. Siddiqi te matrix w If Λ = (λ -,) is an infinite matrix with real entries such that
KX-
belongs to V for all x£ c and a-limAx = limx ,
then Λ is said to be σ-regular [9]. Schaeffer [5] gave necessary and sufficient conditions in order that a matrix be σ-regular.
VreOminavtes. A 27T-periodic function f is said to have bounded p-variation Lf
| 1 / P
U Z ) 0 < P < c o ) or to belong to the c l a s s V , i f
lim s u p i y \f^.)--f(t. ) \ V \
wher
P i - ! e P : 0 = £ . < £ , < £ , , < . . . < £ = 2π i s a p a r t i t i o n of [ 0 , 2 π ] w i th
μ (P) = max £ . — t . , 1 ^^^n '
for an arbitrary ε > 0 . We write simply V for the class of functions of bounded p-variation on [ 0 , 2 π ] , This class was first introduced by Wiener [7] and was further studied by Young [8], We define Fourier-Young coefficients (cf. Siddiqi [6]) of f£V ( 1 < p < » ) by
2π m P
/(n) = (2ΤΓ)-1 [ e~int dft) (n = 0 , ± 1 , ±2 , . . . ) , b
We denote the sequence oo
Akix) = \ f m ^ k x + / ( - & ) e"zkx - π " 1 V <*(* .) cos k x-x . )
«7 = 0 where £?(#·) denotes the jump of f a t # . € [ 0 , 2 τ τ ] . Recen t ly we [6] genera
te ΰ l i z e d a c l a s s i c a l theorem of F e j e r [2] i n the fo l lowing form.
THEOREM A. If / 6 7 ( 1 < p < oo) , then the sequence
yi(b cosnx — a sinnx)>
is summable (N , p ) to π _ 1 d(x) , p rovided t h a t (N ,p) i s r e g u l a r and
< » i H ' S l - ( * ) ™ ί ^-»(Pf) fc= l £:= l
where a and £> are Fourier cosine and sine coefficients of / respectively.
Main Results. Now we generalize theorem A by using σ-regular matrices and we prove the following theorem.
Sequences of Fourier-Young Coefficients 153
THEOREM 1. Let Λ = (λη ,k) be a σ - r e g u l a r m a t r i x . Then for every x£[0,2i\]
and for every f€ V ( 1 < p < <») we have
V oo
lim ^ 4 T Y Y Haü W , k) Ak(x) = 0 v j = o k = i
uniformly in n, if and only if v °°
lim ^j-j- ^ \ \öü\n),k) cos kt =0 J = 0 k = 1
uniformly in η and for all ϋφ 0 (mod 2π) , where σ (η) = σ (σ ( . . . )σ (η) ) is operated j times.
PROOF. We can write (cf. Siddiqi [6]) 2π
ΑΛχ) = τΓ1 [ cosk(x-t) dh(t) , where
CO
Ht) = fit) - τΓ1 £ d( .) <7(t-x.) , J = 0
and #(*) = ^ (0< t < 2π), gr(O) = (2π) = 0,
outside of [0,2π], g is defined by periodicity. Therefore V oo 2 ΤΓ
~ J £ £ A ( c J ( n ) , k ) ^ ( * ) = TT-i J X v n ( x - t ) c?/2(t)
where V oo Kvn(t) = ^ Τ Σ Σ A ( ° J ( n ) ^ ) coskt .
J = 0 fc=l Hence we have to show t h a t
lim f K (x-t)dh(t) = 0 I Vft . . .
V + oo 0
uniformly in n, which is equivalent to show that 2π
lim Kvnx-t)dht) = 0 (1)
uniformly in n for every /€ V (l < p < °°) and for every a: € [0,2π] , where 0 < δ < 2ττ.
Now, Λ being σ-regular, i.e. for all x£c, Ax € 7 , therefore ^ac&oo> and AX € I . Hence there exists a constant M such that for all n
oo co v
sup ( \ ^ 1 £ X(oJ'(n),fc)|)< M . (2)
154
Therefore
\K x-t)
R. N. Siddiqi
^77 y \(oü\n),k) cos kx-t) (3) j=o fc=i
< M
k=i j = o for all V and for all n follows from (2). But, by a well known theorem on weak convergence of sequences is Banach space of all continuous functions, it follows (cf. Banach [1]) that (1) holds if and only if
\K (x-t) I <M
and lim 77 y y λοΰ\η) ,k) cos kt =0
J=0 k=l uniformly in n for all ϋφθ (mod 2π) , which are true due to (3) and the
hypothesis of the theorem. This completes the proof of Theorem 1.
Now we define the sequences oo
iBAx) = f(k) e ' l k x - f ( - k ) e ~ ^ X - iiT1 S d(x .) sinkx-x.) K- L-t J J
and \kx) ± %Bkix) = 2 / (±fc) e
±ikx i v JI„ \ J^ikix-xj.) ϊπ-1 I d(x.) e "g
and we, similarly, can prove the following;
THEOREM 2. Let Λ = (λ -,) be a σ-regular matrix. Then for every x G [0,2π] and for every / G V ( 1 < p < °°)
V oo
(i) _L· V V lim ^j-j- ) ) \oJ(n),k) Bk(x) = 0 j=o k=i
uniformly i n n , i f and only i f
V oo
l i m ^ j y y λοΰ'(n) ,k) sinkt =0
j = o k=i
uniformly i n ft for a l l t^O (mod 2π) , and
v oo
(ii) l i m d r Σ Σ Mc^'C").*) Ufe(«) ± iBk(x)) = 0
uniformly i n n , i f and only i f
l im J5T Σ Σ λ<σί'<»>·*) t k t
= o J = 0 k = l
Sequences of Fourier-Young Coefficients 155 uniformly in n for all t Ψ 0 (mod 2π).
Summability of moduli of allied sequences. In connection with Theorem 1, we may ask whether the sequences of moduli is also summable Λ (or F.) to zero. More generally, we may ask whether the moduli of the sequences A-r(x)
and, B,(x) are summable Λ to zero. We answer this question affirmatively in the following theorem.
THEOREM 3. Let Λ = (λ ν) be a σ-regular matrix such that Ύϊ , Κ _________________________________
l i m ^ - y Y y X(oC' (n) 9k) coskt = 0 v j = o k=i
uniformly i n n for a l l ϋΦΟ (mod 2 π ) , then
(4)
2 = 0
2 = 0
( i ) lim^TT \ V λοΰ (n)9k)\Ak(x) v L L '
j = o k=i and a l s o
v °° ( i i ) l i m V T T X 1 A ( a J ' ( n ) , k ) J Z ^ U )
V j=0 fe=i both uniformly in n.
PROOF, (i) Since (λ J is a σ-regular matrix and (4) holds, hence from theorem 1 it follows that
2π y oo lim [ ~ ) ' y XoJ (n),k) cosk(u-t) h(u) = 0 (5)
0 J = U / C = l
and a l s o 2π 32, ~
lim [ — ~ ^ 2, λοΰ' (n) ,k) cosk(u + t -2 x) dh(u) = 0 . (6) v o j = o fc=i
If we denote OTr
< ( * , * ) = ( 2 π ) " ΐ | ^ -y - y y Ä ( a J ' ( n ) , l ( ) c o s M u - t ) +
0 J = l /c=0
and 2π
+ cos ku + t - 2x)\dh(u)
Kcn (x,t) = ( 2 π ) * [ vTT I Z X ( a J ' ( n ) , ^ ) | c o s / c ( u - t ) +
b j = 0 k _ i
-cos fe (w + t — 2a: ) j 6?/ZM
156 then
and
because
and
R. N. Siddiqi 2π
L n><-
L X k = o nk
Ak(x)
,(oo)
= τ\~ι f K° (t,x) dh(t) ,
2π π"1 j i^ t,x)dh(t)
o 2
2π 4 Gc) = π_1 j cosk(x-t) dh(t)
0 2π
ß.(aj) = π"1 sink(#-£) dh(t)
follows from decomposition of / such that
Ht) = f(t) - π"1 ; d(x ) gt-x.) and g(t)
J = 0 is defined in Theorem 1. It follows from (5) and (6) that
lim K° t,x) lim K° (t9x)=0 n^ °° 1 n-***- 2
for all t and for all x. By an argument similar to that used in the proof
of Theorem 1, we can show that 2π [ K° (t9x) dh(t) 0 '
tends to zero uniformly in n and for all t Φ 0 (mod 2π). This proves part
(i) of Theorem 3. Similarly we can prove part (ii) of Theorem 3.
Similarly we can prove the following:
THEOREM 4. Let Λ = (λ , ) be a σ-regular matrix such that
V oo
l i m — y ) > λ (od (n) ,k) e = Q
J=0 fe=l uniformly in n and for all t Φ 0 (mod 2π), then
lim ^-p ) £ A(aJ (n),k)|^(*) + i^(*) J = 0 fc=l
uniformly in w.
If we denol rem from Theorem 3 by applying Schwarz's inequality
If we denote |Λ| = ( |λ -, \ ), we can easily deduce the following theo-
S e q u e n c e s of Fourier-Young Coefficients 157
THEOREM 5 . Let_ Λ = (λ ) be a σ - r e g u l a r m a t r i x such t h a t
l im ^ γ y V XoJ ( n ) ,k) cos kt = 0 v j = o k=i
uniformly in n for all tj=0 (mod 2 π ) , then
(i)
and also
both uni formly in n .
«7 = 0 k = l
V oo
lim^y y y |λ (ö*1 (n),k) Bk(x) V j ^ O k=l
= 0
If we choose a m a t r i x
for /c = 0 , 1 , 2 , . . . n and λ n, fc
n,k n+\
= 0 for k = n+\ , n+2 , . then we obtain the method of arithmetic mean which clearly satisfies the two hypothesis of Theorem 5 and we obtain the following extended version of Fejers theorem [2].
THEOREM 6 . If / £ V' ( 1 < p <oo) and x € [ 0 , 2 π ] , then n + v
l im n -> °°
im^TT Σ U f e ( x ) |
uniformly i n V = 0 , l , 2 , . . . . fc = v
REFERENCES
1. S. Banach, Theorie des operations lineares, Warsaw, 1932. 2. L. Fejer, Über die Bestimmung des springes einer Funktionen aus ihrer
Fourierreihe, J. Reine Angew Math. 142 (1913), 165-168. 3. G.G. Lorentz, A contribution to the theory of divergent sequences,
Aota. Math. 80 (1948), 167-190. 4. R.A. Raimi, Invariant means and Invariant matrix methods of summability,
Duke Math. J. 30 (1963), 81-94. 5. P. Schaefer, Infinite matrices and Invariance means, Proc. Amer. Math.
Soc. 36 (1972), 104-1 10. 6. R.N. Siddiqi, Determination of the jump of a function of Wiener's class,
J. Univ. Kuwait (Sei.) 4 (1977), 15-24. 7. N. Wiener, The quadratic variation of a function and its Fourier coeffi
cients, Massachusetts J. Math. 3 (1924), 72-94. 8. L.C. Young, An inequaltiy of Holder's type connected with Stieltjes
integration, Aota. Math. 67 (1936), 251-282. 9. A. Zygmund, Trigonometric Series, Vol. I, Cambridge, (1959).
MULTIPLIERS AND ISOMETRIES OF ORLICZ SPACES
W. DEEB Department of Mathematics, Kuwait University, Kuwait
1. Introduction. Let φ be a strictly increasing continuous subadditive function defined on [0,°o) and satisfy φ(χ) = 0 if and only if x=0. Such a function is called a modulus function. Let (X,F,y) be a measure space, and suppose \i(X) < °° . The Orlicz space L. is the set of all complex--valued measurable functions / defined on X satisfying
/ H\fM\)d\i(x) <». X
We define a metric | | on L. by
Ι/Ι φ= i<S>(\f(.x)\)dv,
with the metric (L , | |,) becomes a complete linear topological vector space. If φ(χ) = x , 0 < p < 1, then L, is the space Lr and if φ is bounded, L, becomes the space of all measurable functions.
Using the fact that φ is subadditive and increasing, one can easily show that , f v
1 im < φ (1) x
X ->» from which it follows that L1 c: L, for all φ modulus. If φ(#) = log (1 + ar ), 0 < p < 1 , then φ is modulus and the corresponding L, space will be denoted (for obvious reasons) by N . See [6] for more details.
A multiplier of L, is a measurable function g on X which satisfies fmg € L. for all f£L,. The set of multipliers of L, will be denoted by M(L±). A linear map TiL^-^L, is called an isometry if it satisfies |Τ(/)|, = |/L for all f^-L. . If g is a measurable function on X such that the map m defined on L by m (f) =f*g, is an isometry then m is called multiplication isometry.
159
160 W. Deeb In this paper we will characterize M(L.) for a class of modulus func
tions. We will also find the isometries of L, and show that m is an Φ g
isometry on L, if and only if \g(x)| = 1 a.e. an X. In what follows we assume that φ is a modulus function which is not bounded and φ(1) = 1 , (there is no loss of generality in doing so), 2. Multipliers of LA THEOREM 1 . Suppose φ satisfies φχ* y) >φ(χ) φ(ι/) for_ x > 1 and y > 0. Then Μ(Εφ) = L°°.
PROOF. Since the function f(x) = \ for all x€X is in L, , then M(L.)cL and it is clear that L <=A/(L ). NOW if g€M(L.) then for f £ L1 we have
SH\g\)-\f\ = / Φ ( Μ ) Φ ( * ) , where
h - φ - Μ Ι / ! ) · But
J Φ(Λ) - J l / l < -hence h£L . Let E = <c : |gOc) | > 1 ,
Φ(ΙβΊ) Φ(Α) = ί Φ ( Μ ) Φ ( Ό + ( Φ(ΙβΊ)Φ(Λ) Ε Χ'^Ε
J φ(ΙοΊΛ) + [ φ(Λ)
< Ι ^ | φ + μ | φ < » .
So Φ(|οΊ) is a multiplier of L1 , hence φ ( | | ) C i which implies g£L , and that proves the theorem.
COROLLARY 1 . MLP) = L°° for all 0 < p < oo .
PROOF. This result is known for p > 1 and the theorem establishes the result for p < 1 .
THEOREM 2. _If_ φ satisfies <b(xy) < Φ(#) + Φ(#) for all x and zy, then M <V = ' Φ · PROOF. As mentioned in Theorem 1 we have M(L J C L , now if f£L, then
x φ' φ Φ J Φ ( | / ? | ) < / Φ(Ι/1) + J Φ(|<?1)
= Ι/Ιφ + Μ φ f o r a 1 1 ? ε £φ
hence ^ € MLA .
COROLLARY 2. MN ) = N for 0 < p < 1 .
Multipliers and Isometries of Orlicz Spaces 161
PROOF. One has only to check t h a t φ(χ) - log (l +x ) s a t i s f i e s the c o n d i t i o n of Theorem 2, and t h a t fo l lows from
( ! + * p ) ( l + / ) > \+(xyf .
3. Multiplication Isometries of L. . The multiplication isometries of H^ are known [3] and for more general spaces can be found in [1]. The following result follows the same lines.
THEOREM 3. Let m :L.->L. be defined by m (f) = f'g where g £ L° . Then m is an isometry if and only if |^(x)| = 1 a.e.
PROOF. If |<7(a;) | = 1 a.e. then m would clearly by an isometry. So suppose that m is an isometry. It is clear that \g | , = |^L for all positive integers n.
Let E = x : \g(x) | > l , then
Μφ - Ι / Ί Φ = | Φ ( ! / Ί ) + I H\gn\), E X^E
J Φ(Ιί?Ί) < Ι<Ηφ for all » = 1 , 2
ff X^E hence
E so by Fatou's lemma we get
I " im<M \gn\ E
and this cannot happen unless VE) = 0 , so |^(x) | < 1 , a.e. Now let A = x : \g(x)\ < l . Again
Similarly
But \g
3 Ιφ
ΐφ
= | Φ ( Ι / A
= ( φ ( Ι / A
= J H\g\ A
l\)
Ί)
)
X^A Φ(\9?
+ v(X^A) .
4- μ (Χ^Λ) .
Ί)
| Φ ( Ι / Ί ) = Φ ( Μ ) >
since \g \ < \g\ on E, we conclude that yC4) = 0 or ^(x)=0 for all x£A.
If the case is the second one, let f=X.9 the characteristic function of A, then
V(A) = J Φ(|/|) = I * ( \ f g \ ) = o, from this we conclude that |#(#)| = 1 a.e.
162 W. Deeb 4. Isometries of L, . The isometries of L , 1 < p < °o , where found by Rudin [5], and for 0 < p < 1 were found by Lamperti [4]. In fact Lamperti found the isometries of L, when φ (/χ) is convex or concave. We will show that his result and method of proof works for φ modulus.
LEMMA 1 . If φ is a modulus function,then
φ(|* + 2/|) + 4>\x-y\) <2φ(|χ|) + 2φ(|ζ/|) for all x , y real. Equality holds if and only if x ,y = 0 .
PROOF. By the subadditivity of φ we have
Φ(Ι* + 2/Ι) < Φ ( Μ + \y\) < Φ ( Μ ) + Φ ( Μ ) and
Φ ( Ι * - 2 / Ι ) < Φ ( Μ + \y\) < Φ( W ) + Φ ( Μ ) hence
φ(|α:-ζ/|) + $\x-y\) <2φ(|*|) + 2φ(|ζ/|) .
If x. y = 0 then the equality is clear. Now suppose that equality holds, since
4>(\x + y\) < φ ( | χ | ) + φ ( | ^ | ) and
<$>\x-y\) < φ ( | χ | ) + φ\χ\) , we must have
Φ(Ι* + 2/Ι) = Φ ( Ι * - 2 / Ι ) = Φ ( Η + \y\) but φ is strictly increasing so
\x + y\ = \x-y\ = \x\ + \y\
hence x. y = 0 . Before we prove the final theorems we need a definition. A map T of
F into itself is called a regular set isomorphism if it is defined modulus sets of measure zero and satisfy
T(X^A) = TX)^TA) ,
if A. are pair wise disjoint and μ(Τ\^)=0 if and only if \i(A)=0. A regular set isomorphism T induces a linear transformation T on the set of measurable functions denoted by T also and is characterized by T(X.) -X ,. \ .
THEOREM 4. Let L be an isometry of L. , then there exists a regular set isomorphism T and a function h such that
L(f) = h.T (f) for all f £ L. ,
Multipliers and Isometries of Orlicz Spaces 163
if a measure μ* i s def ined by μ (A) =\iT~1(A)), then
\Hx)\ = φ"1 ( ^ a . e . on T(X) .
PROOF. Suppose L i s an i somet ry and de f ine
T(A) = x: (L(XA))(x) Φ 0 .
If A and B a r e d i s j o i n t s e t s then
But L i s an i some t ry , so
\L(XA)+L(XB)\^ \ΐ(ΧΑ)-φΒ)\φ= 2\ΐ(ΧΑ)\φ+2\ΐ(ΧΒ)\φ ,
which implies by Lemma 1 that LXA) -L(XB)(x) = 0 a.e.
hence \i[T[A) nT(S)) = 0 so T(A US) = T(^) UT(S) .
One can easily extend this result to any sequence of disjoint sets. Now since
X = X^A) UA
we get TX)^TA) = T(X^A).
Finally, y(Z^i4)= 0 if and only if \iA) = 0 and follows from the definition of T(A), and this proves that T is a regular set isomorphism.
Let /z = T(l), for any set A € F we have \=X +X hence
hx) = (L(X.))(x) + (L(Xy^A))(x)
and that gives hx) = (L[X ))[x) a.e. on TA)
L(XA) = h-XTA) = h-T(XA) .
Now extend this by the linearity of L to simple functions and by the density of simple functions in L, and the continuity of L extend the above result to L, . Finally
Φ μ(Λ) = J φ^,) = l*J = Ι^)Ιφ
= J t(\hxTM\)dv = | H\h\)dv.
But y*(i4) = μ(Τ M^))» so V1 is absolutely continuous with respect to μ , hence < i *
μ(4) =V*(T(A)) = j ^ d y T(i4)
and by the uniqueness of Radon-Nikodym derivative we obtain
164 W. D e e b
Φ(Α) d\i d\i
which completes the proof of the theorem. The next theorem shows that if we impose an additional condition on
then the converse of Theorem 4 will be true.
THEOREM 5. Suppose φ satisfies $(xy) < φ(χ) φ(ζ/) for all x,y in [0,<») then for any regular set isomorphism T and for h which satisfy
ψ(Ν) dy* d\i a.e. where \1*A) = y^"1 A))
we have L(f) =h-T(f) is an isometry on L. .
PROOF. The linearity of L follows from that of T . Let AEF then
- [ T(A)
= V(A)
9 =
Φ( |Λ | ) dV =
= ι*Λ · n \ a.XA
i = 1
V*(T(A))
If
is a simple function with A . f\ A . = φ for all ίφΰ, ^ ΰ
n | £ ( Ο - ) Ι Φ = ΐΛ-ϊ'ίο'ίΙφ = I * · \ ai^xA.)\
h' 1 αίΧΠΑ^\ i = 1
J(l· Σ ei^,)l) d\i
^ = l n n
= J j Φ(|Α^|)<Ζμ = X Φ(Ι^Ι) J Φ(|Α|) i = l τ(^) i = i T ( V Y φ(|α.|) μ(4.) = \g\. .
The rest of the proof is straight forward.
Multipliers and Isometries of Orlicz Spaces 165
REFERENCES
1 . W. Deeb, R. Khalil and M. Marzuq, Multiplication isometries of Hardy-Orlicz spaces, (to appear).
2. F. Forelli, The isometries of HP , Canad. J. Math. 16 (1964), 721-728.. 3. M. Hasumi and L. Rubel, Multiplication isometries of Hardy spaces and
of double Hardy spaces, Eakkaido Math. J. 10 (1981), 221-240. 4. J. Lamperti, On the isometries of certain function-spaces, Pacific J.
Math. 8 (1958), 459-463. 5. W. Rudin, L^-isometries and equimeasurability, Indiana Univ. Math. J.
25 (1976), 215-228. 6. L. Waelbrock, Topological vector spaces, Lecture Notes in Mathematics,
Springer-Verlag, No. 331 (1972), 1-40.
CONE ABSOLUTELY (p,g) — SUMMING OPERATORS FROM BANACH LATTICES TO
BANACH SPACES AND THEIR ADJOINTS
K. I. KOSHY* Department of Mathematics, Al-Fateh University,
P.O. Box 13323, Tripoli, Libya
Abstract The concept of absolutely (p,q) — summing operators between Banach spaces is extended to Banach lattices. The Banach lattice version of "Banach-Goldstine Theorem" is proved.
The new classes of operators are introduced here through certain Banach lattice valued and Banach space valued sequence spaces. Some results from the Theory of Tensor Products are also used.
1. Introduction. Notations: Notations and terminology are followed from [6]. E and F are used for real Banach lattices; G and H for real Banach spaces. All operators considered are linear. Tl stands for the adjoint of the operator T.
£r (EyF) is the class of continuous, regular operators from E to F . p1 is the index conjugate to p. That is 1/p+l/p' = 1.
j> Preliminaries [6]. If E and F are Banach lattices, £ (E,F) is Banach space w.r.t. the norm IITil. defined by \\T\\ : = inf ||Sll : ± T < S] for T££ (E,F). If, in addition, F is order complete, £ (E,F) becomes a Banach lattice.
If xi , x' , . . . , x' are order bounded linear forms on E, their supre-mum is given by n
(sup x'i)x) = sup ^ (χ^,χ^) , τ=ι
where the right hand side supremum is taken over all positive decompositions of x € E such that x = χΛ + xn + . . . + x . + 1 2 n
In addition to the classical ε- and π-tensor products, we make use of the |ε|- and |π| -tensor products also. The |e|-norm is the lattice norm 'Current address: Kachirapadinjattel, Oonnukal, P.O. (Via) Omallur (Kerala) India.
167
168 K. Koshy
induced on E®F by £r(E', F). The |π|-ηοπιι on E®F i s defined as follows: if u£E®F,
|Μ||π| : = inf V IKIHI^II :xieE+,yieF+ and | M | < Y ^Θζ/^
It is known that the dual of E ® F is norm and order isomorphic to £ E^F1),
Further the inequality ε<|ε|<|π|<π is always satisfied.
2. Banach Space-Valued Sequence Spaces. DEFINITION 2.1. A sequence X=(x^)i from a Banach space G is called absolutely p-summable, (l<p<oo) if (||x.||) . f £ l .
The set of all absolutely p-summable sequences from G is a Banach space w.r.t. the norm ou-, where
a (X) = < Uw] 1/P
, if 1 <p<o
, if p=c
This sequence space is denoted by ^^G
DEFINITION 2.2. A sequence X = (x .) . from G is called weakly p-summable, (ΐ<ρ<οο)χ if for each x1 € G1 , the sequence
v\ t' / ^ ε N p The set of all weakly p-summable sequences from G is a Banach space w.r.t.
Γ the norm ε , where p r ! y \(x x'\
sup I L \\^i » x / Itf'lKl ^ = 1 ^ (X) = sup sup l 'lll i
<*,,*')
, if 1 <p<c
, if p =
This space is denoted by t (G).
It is a well-known fact that for 1 <p<<», there is an isometric isomorphism between 1 (G) and £ (I / , G) and that 1 AG) is isometrically isomorphic to £(cn9G). In both cases an element (#.).-.. or I (G) is associated with the operator T defined by
-> i)- y i=\
where (t.) . , .. is an element of Z , or cn as the case may be. v ^/^ £ N p 0 J
DEFINITION 2.3. A sequence Z= (a .) . c „ from G is called strongly p-summable, if for each X1 = (a;.).^.T from 7- / ( G M the series
Summing Operators from Banach Lattices to Banach Spaces 169
oo
X · , X · ) — l \ Λ , Λ /
l__s \ x » ^l \ ' / converges.
The set of all strongly p-summable sequences from G is a Banach space w.r.t. the norm σ , where
P op(x) = sup | * ,* ' ) | : y (*) < '' This space is denoted by I (G\ .
If we consider G-valued sequence spaces with only a finite number of coordinates; the spaces introduced by the above definitions are denoted respectively by
lnvG\, ln
pG) and 1^(G) .
It has been proved that their normed duals are respectively
lU, <?' , ln lG'\ and f1, (<?') where i + -i-= 1 .
3 . Lattice Summable Sequences from a Banach Lattice. If (x.) . is a sequence from a Banach lattice E, (\x .\) . is absolutely p-summable ( l < p < ° ° ) , if (χ.).-„ is, since ||^·|| = || \x ·| || for every i .
However, this is not so in the case of a weakly p-summable sequence. Hence we formulate the following definition.
DEFINITION 3.1. A sequence X= (x .) . ,7 from a Banach lattice 2? is called t tc/lf
weakly lattice p-summable (l < p < ° ° ) , if ( \x · | ) · fN is weakly p-summable. The set I [E] of all weakly lattice p-summable sequences from E is a
Banach lattice w.r.t. the norm e and the coordinatewise ordering, where e (X) = ε„((|*.|).)
= sup [l |< |* . | ,* ' ) | ρ ] 1 ρ : |μ ' | |< >. * ' " ' As Ε7 is also a Banach lattice, it is enough to take the supremum above over the positive part of the unit ball of E1 . It is evident that ε (X) <e (X)
and if X >0 , ep(Z) = ep(J) . In the following Theorem, I [E] is identified with certain space of
operators. We recall that £r(E ,F) is the class of continuous operators between the Banach lattices E and F that can be expressed as the difference of two positive operators (Ref. [6]).
170 K. Koshy THEOREM 3.1. Let E be a Banach lattice. For 1 < p < °° 9 I [E] is isometric-ally lattice isomorphic to £r(l f , E) and I [E] is lattice isomorphic to ^(cQ,E).
This theorem shows in particular that the spaces £ (I , E) , 1 < p < °° and £r(c„,E) are Banach lattices whether or not E is order complete.
DEFINITION 3.2. A sequence X= (x .) .__7 from a Banach lattice £ is called strongly lattice p-summable ( 1 < p < °°) , if for each X1 = (xf.) ... in I , [Ε' ] , the series
l <ν*ί> 7 = 1
converges.
THEOREM 3.2. The set 7 ]£[ of all strongly lattice p-summable sequences from a Banach lattice Ef equipped with the norm s and the coordinatewise ordering is a Banach lattice, where
sp(X) := sup |(*,*')| :ep, (X' ) < l .
PROOF. We first prove that s is finite. If X = (x .)°°. . £ I ]E[ , we define r p v ^/^=1 p
a linear form F on I / [Ε'] by p *(*') = ). (^Ο' L.
i=l Obviously F is the pointwise limit of the continuous linear forms F (n €. N)
on 7 ΛΕ'] defined by p n
n K K ^ / ^ = l z_: x ^ ^ 7
^=1 and therefore F is continuous. Obviously s X) = \\F\\ and hence is finite.
The norm axioms can be easily verified for s . The routine proof of the norm-completeness of 1 ]E[ is omitted here.
Now we show that s is a lattice norm on I ]E[ . For this, we have to show that s (x) < s (j) if 0<X<Y and s (|j|)= s (X) for arbitrary XyY from I ]E[ . The first of these is immediate from the definition of s . To prove the second, we observe the following result on order complete vector Lattices.
If / and g are positive, lattice disjoint and order continuous linear forms on an order complete vector lattice E each element x£E can be split into two positive, disjoint components u and V such that x=u+V and f(u) =g(v) =0. (Ref. [6], pp. 78-79).
Without loss of generality, we can suppose that s (X) = 1 . If possible, let s \x\) Φ s (X) . This means that s (\x\) > 1. Hence there is a positive sequence (x*. ) . in the unit ball of 7 ,[E/] and a natural number n
Summing Operators from Banach Lattices to Banach Spaces 171
such that
\ ( \ x i \ , x ' i ) > l. i=l
+ . .. + Let x . = x . - x . . Since E can be identified with a sub-lattice of E" , x. and xT are order continuous linear forms on the order complete Banach lattice E1. Therefore x1. can be decomposed as x1. = ur. + V1. where u'. >0 and v'.>0m
u'. h vl. = 0 and /# . , u'.) = (x . ,t; ί\ = 0 . Now construct a sequence (yJ.) . by setting ty7.= u'.-V1. , for t = 1,2,
...,n and (z/ ) =0 for i>n. Clearly (Ϊ/·)·. is an element of the unit ball of Z [£'], as e ((z,'.)" Ί ) = e ((a;7. )* \ ) < e ((*'.)" η . But
cxj n n y (*·>/ ·) = y (*.,2/;Λ = y d * . i , * ' Λ > i . t=1 i=1 -£= 1
This contradicts the assumption that s (X) = 1 . The proof of Theorem 3.2 is now complete.
It is clear that the various sequence spaces introduced above satisfy the relation I (E) C I ]E[ c I E C I [E] c I (E) and σ > S > a > e > ε .
p \ f - p — pL J ~ p - p v / p p p p p The tensor product X ®F is a subspace of each of these sequence spaces
and it is we 11-known that I (F) , I [F] , I ]F[ and I (E) induce respectively the ε-norm , the |e|-norm, the |ττ|-ηοηη and the π-norm on I ® F.
Next, we find the normed duals of I [F] and 1 ]F[ . We need the following Theorem on Banach lattices which is the extension of Goldstine's Theorem on Banach spaces. THEOREM 3.3. Let F be a Banach space Fl and FN be its normed dual and bidual respectively. Let F be ordered by the positive cone F and F ' , F/;
be ordered in the natural way. If U, U ' , U " , respectively denote the unit balls of F , F; , Fn , respectively then i/+ : = ί/ Π F+ is σ (Fn, F ; ) ~ dense in U" : = U" Π F" . + + PROOF. Let fo' denote polar w.r.t. the duality (F , F'. Then (F+)°= - (F') + and (-F ) = (F') . Now, U and F both are convex and norm-closed and con-tain (9. Therefore (U+)° is the σ (F', F)-closed convex hull of U° U (F )°
which is the set U + (F ) . Now, if * denotes the polar w.r.t. the duality (F' , F") , (U°+(F+)0)* is equal to U°* Π (F+)°* . Thus
(U+)°* = if* Π (f+)°* = U" Π (ί·")+ = (U")+ .
The theorem follows immediately by an application of the bipolar Theorem.
Note: X, \x\ and Y are denoting the sequences (#·)·_ > (Ι ·Ι)·_·ι an<^
172 K. Koshy COROLLARY. Let E be a Banach lattice and E be identified with a sublattice of E" . Then the positive part of the unit bail of lL[E] i_s O(EN ,E ) —
dense in the positive part of the unit ball of I [E ].
THEOREM 3.4. Let E be a Banach lattice and 1 < p < °° . Then (i) (I ]E[) is isometrically lattice isomorphic to I /[El] .
(ii) (Z [£]) is isometrically lattice isomorphic to 1 /]E [ .
PROOF, (i) Since f"\E[ is the tensor product f1 ® E, its dual is £r(l , E7 ) which is isomorphic to I /[S*7].
(ii) We know already that the dual of t^lE] is topologically and order isomorphic to I /]E/[. In fact this isomprphism is given by
( ^ ] ) ' 3 f - > ( < ) " = 1
where (x. , xl λ = ((δ . .x) . , , f ) > ό" . . being the Kronecker delta. It remains to show that this isomorphism is an isometry. That is, we
have to show II f II = Siiix1.) . H) . Or we have to show that p 1 t= ί
n
supifH (x.,x.)\ : 0 < [x.)n. , e ((a;.)" J < »1
This is true because of Theorem 3.3.
COROLLARY. It E is a Banach lattice and 1 < p < oo ,
ln\EV' = ln\.E" ] and ( Z n M ) " = Zn]Z?"[ .
THEOREM 3.5. Let # be a Banach lattice. If (x.) -=1 is an element from E ,
then n
X, , = X and s ((a; .) . , ) = sup I x . I τ'τ-ι II ^ ' t ' l l - t ι - ι » 1 < i < M *'
^ = 1
PROOF. The assertion on e is quite easy to prove. The second assertion follows from the definition of the supremum of a finite number linear functional on E together with the fact that E is a sublattice of En .
4. Cone Absolutely (p,q) — Summing and (p,q) — Majorising Operators. DEFINLTION 4.1. Let E be a Banach Lattice and G be a Banach Space; p,q
be real numbers such that 1 < p, q < °° . An operator T: 2? —>- G is called cone absolutely (p,q) —summing, if there exists a real constant 3 such that for any finite sequence (x ,x2 , ... , x ) from E the inequality a„((Tx.)n. ) < ße (% ·)η· ) is satisfied.
Summing Operators from Banach Lattices to Banach Spaces 173
DEFINITION 4 . 2 . T : G -^ E i s c a l l e d ( p , q ) - m a j o r i s i n g ( l < p , q < o o ) i f
t h e r e e x i s t s a c o n s t a n t 3 such t h a t fo r any sequence (x 9x , . . . , x )
from G9 t he i n e q u a l i t y s ((Tx .) ._ ) < 3 α ((x .) ._ ) i s s a t i s f i e d . P t'L· — I £7 is υ — 1
DEFINITION 4.3. T : E -^ F is called perfectly (p , q ) - summing < 1 < p , q < ~ ) if there exists a constant $ such that for any finite sequence x,, x_, . . . , xn) from £ the inequality s (( a:.)· < β δ σ ((^·)·_ι) is satisfied. The set of all cone absolutely (p9q) — summing operators from E to G is denoted by K (E,G) . The set of all (p , q)—majorising operators from G to E is denoted by M (G9E).
p,q The set of all perfectly (p, g) — summing operators from E to F is denoted by P (E ,F) . The infimum of 3 in the definitions 4.1, 4.2, 4.3 are respectively called the cone absolutely (p9q) —summing norm, (p, q) — majorising norm, and perfectly (p, q) — summing norm of T and are denoted by
IMI* ' WTh and INI? · If T:G-^H9 let ^ : Gn -^ //n be defined by T j . x ^ χ) = ( ^ ) ^ = χ
Then T€ X (£,£) if and only if T : Z"[F] -> ZnG is boundedly conti-p ,q ^ n H p l J ^
nuous. (That is, T is continuous and || T || is bounded for each n E N). Similarly, T€M (G,£) if and only if T : ln G-> ln ]E[ is boundedly J * p ,q J n β l J p ^ continuous and T € P (#,F) if and only if T : f1 [E] -> ln ]F[ is boundedly
p,q J n q p J
continuous. The duality relations among the various sequence spaces introduced here
enable us to prove the following. THEOREM 4.1. Let E, F be Banach lattices and G be a Banach space and 1 < p , q < oo . Then,
( i ) TEKp q(E,G) iff . T1 £ M,9 i G',E')
( i i ) TeMp'9q(G9E) i f f T'£Kq,99p,E'9G')
(iii) Tep^t^F) iff. τ' e pq/9 p / ( Ρ ' , Ζ ' ) .
In each case the norm of T equa l s t h a t of T1 .
COROLLARY. ( i ) TEK (E9G) i f f TnEK [E"9 G")
p,qK ' ' P,q ( i i ) TEM (G9E) i f f THEM [G"9E") p9qK p9qK
(iii) TEP (E9F) iff T1'EP E" , FN) p,q — p>q
It is obvious from the definition 4.1 that TEK E9G) implies that P»<7
TE£(E9G) and ||Τ||< \\Τ\\ν · Α similar remark holds for the other two P > <?
174 K. Koshy classes as well. If p<q, each of these classes contains only the zero operator. Further a perfectly (p,q) — summing operator is both cone absolutely (p , q) — summing and (p , q) — majorising. Besides, K ( E 9 G ) = £ ( E , G )
and M (G,E)=£(G,E). It can be proved that the spaces K ( E , G ) , p,1v ' x ' v v p , q K '
M (G,E) and P (E,F) are Banach spaces. p , q K p , q x
REFERENCES
1 . H. Apiola, Duality between spaces of p-summable sequences, (p,q) — summing operators and characterisations of nuclearity, Math. Ann. 219 (1976), 53-64.
2. J.S. Cohen, Absolutely p-summing, p-nuclear operators and their conjugates. Math. Ann. 201 (1973), 177-200.
3. K.I. Koshy, Cone absolutely (p, q) — summing operators from Banach lattices to Banach spaces and their adjoints. Dissertation, Universitaet Dortmund, West Germany (1976-77).
4. A. Persson, A. Pietsch. p-nucleare und p-integrale Abbildungen in Banachraeumen. Stud. Math. 33 (1969), 19-62.
5. A. Pietsch, Absolute p-summierende Abbildungen in normierten Raeumen. Stud. Math. , 28 (1967), 333-353.
6. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, 1974.
THE STABILITY OF LINEAR MAPPINGS AND SOME PROBLEMS ON ISOMETRIES*
T. M. RASSIAS Department of Mathematics, University of La Verne,
P.O. Box 51105, Kifissia, Athens, Greece
Abstract, In this lecture, I will try to give a brief account of some of the main entries of development (with a few new research problems) in the very active subjects of the stability of linear mappings in Banach spaces and iso-metries between metric spaces.
1 . The Stability of Linear Mappings in Banach Spaces.. In 1940, S.M. Ulam gave a lecture before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among these was the following question concerning the stability of homomorphisms. We are given a group GL and a metric group G^ with metric <£(.,.). Given ε>0 does there exist a δ>0 such that if f'.G —> G
satisfies d(f(xy),f(x)f(y))<& for all x,y in G, , then a homomorphism h\Gx -> G2 exists with df(x),h(x))<e for all x£G (D.H. Hyers [8]).
In 1941 D.H. Hyers [7] considered the case of approximately additive
mappings /:E\ -» E where, E^ and E are Banach spaces and / satisfies \\fx + y)-fx)-f(y)\\ < ε CD
for all x,y in E,. It was shown that the limit
L(x) = lim J-^-~^- (2) n -*«> 2
exists for all xEE-^ and that L:El—>E2 is the unique additive mapping
It is my pleasure to express my gratitude to the organizing committee of the Conference on Mathematical Analysis and its Applications, Department of Mathematics, Kuwait University, for inviting me to deliver this lecture in Kuwait.
175
176 Τ. Μ. Rassias satisfying II fix) — L(x)\\ = ε. No continuity conditions are required for this result, but if fitx) is continuous in the real variable t for each fixed x,
then L is linear, and if f is continuous at a single point of E then L is also continuous. Hyers established (2) in a very clever way by showing the inequality
Ι ^ - ' φ Ι <(-£) by induction on the basis of (1), and then showing that f(2 x)/2 is a Cauchy sequence.
In 1978 (cf. [11]) I generalized Hyerfs result in the following theorem:
THEOREM. Consider El9E2 to be two Banach spaces, and let fiE -+E be a mapping such that fitx) is continuous in t for each fixed x . Assume that there exists Θ = 0 and p £ [0,1) such that
\\fix + y)-fix)-fiy)\\ <a c cv 1 L = ^ ^ — > L 1 - ^ — ' ΰ II ^ Θ for any x „ y € ΕΛ . II I I P II I I P l Nr + hV
Then there exists a unique linear mapping T: E —> E such that
(3)
| / ( » ) - Γ ( * ) | | ^ _ 2 θ _ χ ε (4)
llxll P 2 - 2 P
PROOF. Claim t h a t
1 Lf(2n*)-f(*)| n-1 2 <- θ V 9m(p-i)
Hp z 2"'^ " (5)
for any integer ft, and some Θ > 0. The verification of (5) follows by induction on ft. Indeed the case ft = 1 is clear because by the hypothesis we can find Θ , that is greater or equal to zero, and p such that 0 ί ρ ^ 1 with
- έ θ . (6) !UI!P
Assume now t h a t (5) ho lds and we want t o prove i t for the case (ft+ 1 ) . How
ever t h i s i s t r u e because by (5) we o b t a i n
| ^ / ( 2 M . 2 * ) - / ( 2 * ) | n - 1 _J < θ \ 2m(p-i)
t h e r e f o r e 2dK ^
II m=o _ ! f(2n 1x)--f(2x)
I ft+1 JKA x ) ilKLX)\ ft _2 <;θ V 2m(p-l)
By the triangle inequality we obtain
Stability of Linear Mappings
j - J i T / ( 2 M + 1a : ) - / . ( a : ) | s | ^ n - / ( 2 r t + 1
a ! ) - i f ( 2 * ) n
+ | U / ( 2«) - / ( * ) | ^θ Ι Ι χΡ ) 2mP~l) . m= o
Thus - L T / ( 2 " + 1 x ) - f ( x :
m(p-l)
and (5) is valid for any integer n . It follows then that
\ f(2nx)-f(x) 2Θ
2-2?
because V m(p-i)
However, for m > n > 0 $
converges to , as 0 p < 1 . 2-2 P
^Λί-^/Λ) n\\ m-n f(2mx)-f(2?
. 9n(P-l) 2Θ || IIP < 2 r r \\x\\ 2-2*
Therefore lim JL ^ 2 ^ ) — L / ( 2
M x ; = 0 .
But # , as a Banach space, is complete, thus the sequence
Br"2n*> converges. Set
It follows that
Tx) = lim — f2nx) .
/(2"<«-!,))-/(2"I)-r(2''!,)||se(|2"If+i,|2"!,f)
-*Ρ-(ΙΗΜΜΓ) Therefore
\\\f(2n(, + y))-f(2\)-f(2nh)\\^P-^.e.(\\x\\P + \\y\\ P\
lim ±\f(2n(x+y))-f(2"x)-f(2ny -pfnn \\\ <- T nn(p-i) Q (\\ IIP , — I ( 2 .v ) II = 1 im 2 £ · Θ · I \\x\\ +
178 Τ. Μ. Rassias
I lim \ f(2n(x+y))- lim -L f(2nx) - lim \ f(2ny)
n-+oo 2 n-*oo 2 n->oo2
T(x + y) - T (x) -T(y)\\ = 0 for any x , y € Ε
or T(x+y) = Τ(χ) + T(y) for all x,y£E .
Since T(x + y) = T (x) + T(y) for any χ^£Ελ. Τ (rx) = rT (x) for any rational number v. Fix xQ £ E-, and p € E* (the dual space of E ) . Consider the mapping
i? 3 t —> p (T(tx) ) = φ (t) . Then φ :R-+R satisfies the property that φ (a + b) = φ(α) + φ ( & ) . φ is a Borel function, because of the following reasoning. Let
P ( f ( 2 n t x )) φ(£) - lim
ft ->oo 2 and set Λ / «/0ft , Ν λ
Φ (t) = . Then φ (t) are continuous functions. But φ(£) is the pointwise limit of continuous functions, thus φ(£) is a Borel function. It is a known fact that if o : i f ^ i f is a function such that Φ is a group homomorphism, i.e. <\>(x + y) = φ (x) + φ(ζ/) and φ is a measurable function, then φ is continuous.
ft
In fact this statement is also true if we replace R by any separable, locally compact abelian group (cf. W. Rudin [14]). Therefore φ(ϋ) is a continuous function. Let a ER. Then a =lim r , where \r is a sequence of
ft ->oo ft ft
rational numbers. Hence
φ ( α £ ) = φ ( £ lim r ) = lim <\>tv ) = ( l im r ) φ(£) = αφ(£) . ft->00 η ft->00 n ft->C» W
Therefore φ(α£) = α φ ( £ ) for any a€i?. Thus Tfox) = off (a:) for any a ER.
Hence T is a linear mapping. From (7) we obtain
\\^ f(2nx)-f(x)\\ _ nn l im ^ lin
ft->oo II IIP ft->oo 2 - 2 P
or equivalently
\T(X) -f(x)\\ ? n < ε , where ε = — ^ — . (8)
IIP 2 - 2 *
Thus we obtained (4).
Stability of Linear Mappings 179
We want now to prove that T is the unique such linear mapping. Assume that there exists another one, denoted by g : E,-*E2 such that
T(x) 4 g(x) , * € £ 1 #
Then there exists a constant ε , greater or equal to zero, and q such that 0 έ q < 1 with
By the triangle inequality and (8) we obtain
\τ(χ)-9(χ)\\ <|τ(χ)-/(χ)| + |/*)-^)Ι < ε ||χ|Ρ+ e1L|'7
Therefore
| r ( x ) - £ ? ( x ) | = \\±T(nx)-±g(nx)\\
~JT(nx)-g(nx)
i('l nx\\ + e i | n a : | | )
p-i II I*' <?-i || = nc ε m + η^ ε , a:
Thus l im !T(a;) - # ( x ) = 0 for a l l x £ i and hence T(x)=g(x) for a l l a? € E\ . n ->oo II 'I i i
Q . E . D .
This solves a problem posed by S.M. Ulam [15]: When does a linear map
ping near an "approximately linear" mapping exist ? The case p=0 was answered by D.H. Hyers [7]. Therefore we have succeeded here to give a generalized solution of Ulam's problem.
In 1982 H. Drljevic and Z. Mavar [5] following the approach in my paper [11] studied a similar problem on A-orthogonal vectors. In fact they proved the following interesting theorem:
THEOREM. Consider φ to be a functional defined on a complex Hilbert space X and let cp(tx) be continuous in t for each fixed x € X. Assume that there exists Θ ^ Ο and p€[0,l) such that
I I (\ \p/2 I \p/2\ \<px + y) -φχ) -ipy)\^Q l\(Ax9x)\ +\(Ay9y)\ \ (10)
(for x , y €Z for which (Ax, y ) = 0 ) where A is a continuous self adjoint operator from X jm X with the property that dim A (X) =£1,2. Then there exists a unique continuous functional φ which is additive on -orthogonal pairs such that
Up(rc) - φχ (x) * θ ι (Λχ , tf) for any a? € X (11)
where Θ, is a constant.
180 Τ. Μ. Rassias 2. Isometries. We begin with the definition of an isometry : Let X,Y be two metric spaces, d ,d the distances on X and Y. A bisection mapping f i X -* Y of X onto Y , is defined to be an isometry if
d2(f(x) ,f(y))=dl x,y) for all elements x, y of X. If f:X-*Y is an isometry, then the inverse mapping /" l : Y -> X is an isometry of Y onto X. Two metric spaces X and Y
are defined to be isometric if there exists an isometry of X onto Y . It thus follows that an isometry is an isomorphism for the metric space structures. Some of the properties of an isometry are mentioned in [3] and [A]. We now state in which sense an incomplete space can be fattened out to be complete: If (X>d ) is an incomplete metric space, then there exists a complete metric space X so that X is isometric to a dense subset of X . S. Mazur and S. Ulam [10] have proved that every isometry of a normed real vector space onto a normed real vector space is a linear mapping up to translation. Consider then the following condition (distance one preserving property), for f:X->Y.
(DOPP) . Given x , y G X with d (x,y)= 1 . Then d2(f(x) , f(y)) = 1 . This condition was considered by the author in [12]. A.D. Aleksandrov had posed the problem : Under what conditions is a
map of a metric space into itself preserving unit distance an isometry ? Unless the contrary is stated, it is not assumed in the following that
the map is one-to-one, onto, or continuous. Furthermore, it is not even assumed that the map is single-valued.
n n . .~
By E , L will be denoted respectively Euclidean space and Lobacevskn space of dimension n.
A. Guc [6] had proved that a bijective single-valued map f'.L -* L
(n > 2) such that for some number r>0 and each point x£L satisfies f(s"-1(x,r))=s""1(/(x),r) (12)
is a motion, where S (x9r) is the (n-1)-dimensional sphere with centre x and radius r .
THEOREM. (E. Beckman and D. Quarles [2], A. Kuzminyh [9]). Lejt a be a fixed positive real number. A map f:L -* L (n 2) is an isometry if there exists a positive real number b such that from p (x , y ) = a (where xyy €L , ρθ,ι/) is the distance between x and y) it follows that for each pair of points x1 , y1 (x/G/>(^)»^/€:/(i/)),p(^/,z// ) = b . This in particular implies that f is singlej^valued and a = b.
Stability of Linear Mappings 181 THEOREM. (A. Kuzminyh [9]). Let_ DHCEn (n > 2) be an open ball of radius r(Dn) > 1 . There exists a set MCDn such that
1) M has measure zero, 71 71
2) a map f:D -> E is an isometry if, for any point x £ M, from p(x9y) = I y £ D ) it follows that, for any pair x' , y' (xf £ f(x) 9
y'tf(y)), ρ(χ; ,y' ) = i · Remark. It is easy to see that the set M cannot be countable, since a union of a countable set of spheres of radius 1 has measure zero, i.e. does not contain Dn.
E. Beckman and D. Quarles [2] proved that if f:E -> E for 2 ύ n < °° satisfies condition (DOPP) , then / is an isometry. It can be proved that this theorem holds for any n-dimensional hyperbolic space for 2^n<°°; it is true in any n-dimensional elliptic or spherical space if the preserved distance is small enough.
This property does not hold for E , the Euclidean line. A simple counterexample is the following:
Let f'.E -> E be defined by:
ix + 1 if x is an integer point (13)
x otherwise. oo
Also this property does not hold for E , a Hilbert space. A counterexample can be made in the following way:
Let y . be a countable everywhere dense set of points. Define g : Ε°°'-> y . such that d(x 9g (x) ) < \ . Define h : y . -> a . such that h(y.) =* a . with x x x
<*. = (-£-. -£,..., , . . e f (14) ^ \ /2 /2 /2 /
where 6 . . is the Kronecker delta. Then f = gh:E°°-> E°° satisfies condition (DOPP). However / is
not an isometry. It is interesting to examine what happens when the mapping is required
to be continuous. In E , the transformation
f:x-> [x] + x]2 (15) (where [x] denotes the integer part of x and x =x - [x]) is continuous and satisfies condition (DOPP) but is not an isometry. PROBLEM. It is not yet known what happens in E even with the additional condition of continuity on the mapping. My conjecture is that such a mapping,
182 Τ. Μ. Rassias satisfying condition (DOPP), must be an isometry.
A. Beadle [1] has given an example of a mapping in S that preserved the distances ττ/2 and π but was not an isometry. However, this transformation was not continuous. It is an open problem whether or not the distance TT/2 can be preserved by a continuous map which is not an isometry. Also in spherical and elliptic spaces, it is not known whether or not the additional requirement of continuity placed on a mapping which preserves large distances forces that map to be an isometry. Beadle combining continuity and distance preserving properties for the mapping made the following conjecture, which according to the evidence I have now seem to be true.
CONJECTURE. If M is a locally Euclidean manifold of finite dimension greater or equal to two, then there is a distance a such that for any b<a and any mapping f: M-* M, where / preserves distance b implies that / is an isometry.
If f:E -» E preserves some distance, it follows that num. The case 1 = n — m S °° has been discussed above. It remains to examine the case when 1 < n < m < °° ,
In the following we outline a method to show how to construct examples to prove that for each n there exists an m and a unit-distance preserving mapping f:E -» E that is not an isometry. The following example illus-
2 8 . trates the case of a mapping f:E -* E . For this consider partitioning the plane into squares of unit diagonal as follows:
7 8 9 1
3 1 2 3 1
6 4 5 6 4
9 7 8 9 7
1 2 3
where each square contains the bottom edge, the left edge and the bottom left corner but none of the other corners. Now label the nine vertices of the unit
8 8-simplex in E and map each square labelled i to the ^τ:n vertex. This mapping satisfies condition (DOPP) but is not an isometry. Using hexagons instead of squares one can construct such a mapping from E -*E . This idea
extends easily to higher dimension. We will show now that if m is larger than a certain bound φ(η) , then
a mapping f:E -* E ( 1 < n < m < °o) satisfying condition (DOPP) is not necessarily an isometry onto its image.
Stability of Linear Mappings 183
DEFINITION. A s e t QCEn i s c a l l e d " e x c l u s i v e " i f for x,y£Q i t fo l lows t h a t
dx,y) Φ 1 .
yi THEOREM. For each n,E can be partitioned into a finite number of exclusive sets Ω, , Ω0 , . . . , Ω 1 2 ic
ΕΆ = Ω1 U Ω2υ . . . υΩκ ,Ω ΠΩ. = φ for i Φ j . (16)
PROOF (idea). To be more precise let us consider the proof with k=9 for 2 n = 2, the general case is similar. Partition E (as we did before) into a
chessboard pattern of squares numbered from 1 to 9 with period 3 in each coordinate direction.
8
2
5
8
2
9
3
6
9
3
4
7
1
4
7
1
4
5
8
2
5
8
2
5
6
9
3
6
9
3
6
7
1
4
7
1
2
5
8
2
It is clear to see that if the little squares each have diagonal equal, say, to 0.9 (hence sides are > j) then the totality of squares with the number " j " is an exclusive set (j G 1 , 2, . . . , 9 ) (we can place the boundary points of the squares indifferently in one or other of the squares bordering them).
THEOREM. If_ ΕΓ is partitioned into K exclusive sets, then a map Yl K— 1
f : E -* E need not even be injective. K- 1 PROOF. Let z , z , ... , z be κ points in E of mutual distance 1
(vertices of a regular (κ - 1)-simplex). Let now
En = El \)E2 U ... WE
be the above decomposition and define
f(x) = z . where x£E . and j = 1 , 2 , . . . , κ . v d
Then it follows that df(x) , f(y) ) = 1 whenever d(x,y) = 1 since d(x,y) = 1 implies x€E., y€E. for some i^j.
PROBLEM. Find the minimal number κ=κ(η) .
184 Τ. Μ. Rassias REFERENCES
1. A. Beadle, Ph.D. Thesis, Michigan State University, 1977. 2. E.S. Beckman and D.A. Quarles, On isometries of Euclidean spaces, Proc.
Amev. Math. SOG. 9 4 (1953), 810-815. 3. G. Choquet, Topology, Academic Press, New York and London, 1966. 4. J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York
and London, 1969. 5. H. Drljevic and Z. Mavar, About the stability of a functional approxi
mately additive on A-orthogonal vectors, Akad. Nauka Umjet. Bosne i Hercegov. Rad. odjelj. Prirod. Mat. Nauka 69 (1982), 155-172.
6. A.K. Guc, On mappings that preserve a family of sets in Hubert and hyperbolic spaces, Candidate's Dissertation, Novosibirsk, 1973 (Russian).
7. D.H. Hyers, On the stability of the linear functional equation. Proc. Nat. Akad. Sei. U.S.A. 27 (1941), 222-224.
8. D.H. Hyers, The stability of homomorphisms and related topics. Global Analysis-Analysis on Manifolds (ed. Th.M. Rassias), Teubner-Texte zur Mathematik, Leipzig, Band 57, 1983, pp.140-153.
9. A.V. Kuzminyh. On a characteristic property of isometric mappings, Sov. Math. Dokl. 17 (1976), 43-45.
10. S. Mazur and S. Ulam, Sur les transformations isometriques d'espaces vectoriels normes, C.R. Akad. Soi. Paris 194 (1932), 946-948.
11. Th.M. Rassias, On the stability of the linear mapping in Banach Spaces. Proc. Amev. Math. Soc. 72 (1978), 297-300.
12. Th.M.-Rassias, Is a distance one preserving mapping between metric spaces always an isometry? Amev. Math. Monthly 90 (1983), 200.
13. Th.M. Rassias, Foundations of Global Nonlinear Analysis, Teubner-Texte zur Mathematik, Leipzig, Band 86, 1986.
14. W. Rudin, Fourier Analysis on Groups, Interscience Publ., New York, 1962.
15. S.M. Ulam, A collection of Mathematical Problems, Interscience Publ. New York, 1961.
GLOBAL THEORIES OF THE LINEARIZING PROJECTION WITH APPLICATIONS*
W. H. RUCKLEt Department of Mathematical Sciences, Clemson University,
Clemson, SC 29634, USA
1. Introduction. A linearizing operator or projection is a device which converts nonlinear into linear information. The Shapley value, both in the discrete case (Shapley [9]) and the continuous case (Aumann and Shapley [1]) is a well-known example of a linearizing projection. A linearizing projection usually satisfies certain axioms of rationality which assure that it is the "unique, fair" allocation or distribution. This makes it an admirable bookkeeping device because bookkeeping must be linear.
In (Ruckle [7]) a first attempt was made to consider the Aumann-Shapley theory of values in a setting of functionals defined on an arbitrary Banach space E . Unfortunately, except for the space p(Ef) of polynomials of continuous linear functionals on E , the spaces of functionals considered are somewhat contrived. Hence, it is not easy to determine whether a functional encountered in an applied problem is in one of the spaces or not.
In this report we shall describe an effort (Ruckle [8]) to construct three global theories of linearizing projections (called x -values). These theories are called "global" because they refer to spaces of functionals which are defined on the entire Banach space E. Any details which we omit in the present work may be found in the paper cited above, i.e. (Ruckle [8]).
The polynomial theory described in Section 1 is more complete than that given in (Ruckle [7]) for p(E'), not only because it avoids the technical hypothesis that E have dimension greater than three, but more importantly because it applies to polynomials of the form
Invited talk. Partially supported by NSF, DMS 8500946.
185
186 W. H. Ruckle n
G(x) = ^ Fk(x) , (1.1) k=Q
where each F-, is a symmetric multilinear form defined on E. The basic theory of such polynomials is treated in Chapter 26 of (Hille and Phillips [A]). When E has infinite dimension not all polynomials of the form (0-1) are polynomials of linear functionals. For example, the functional T on [0,1] defined by
1 T(x) = | x(t))3 g(t) dt ,
0 where g is a fixed integrable function in a polynomial of type (0-1) but not a polynomial in a linear functional.
It is a short step from a polynomial theory to a theory for entire real analytic functions on E. Such a theory is described in the second section.
The main contribution of (Ruckle [8]) is described in Section 4. It is 1 a theory of linearizing projections defined on the space C (E) of functionals
on E whose first derivatives are (locally uniformly) continuous. This is a natural space of functionals on E, and it is often possible to verify a functional 's membership in the space.
In the final section we describe several sample applications of the mathematical theories constructed.
For E an arbitrary real Banach space, let X be a linear space of real valued functions (functionals) on E which includes the space E of continuous linear functionals on E and has the property that whenever F is in X
and T is a continuous linear mapping from E into E the functional FuT,
defined by F o T(x) = F(T(x)) for x in E, is also in X.
1.1 Definition. Let x be a distinguished nonzero vector in E. An x -value is a function P from X onto E" which satisfies the following conditions:
(L ) P is linear: i.e. P (au + bv) = aPu + bPv ; a, b in R; u , v in I;
(L ) P is x -symmetric; i.e., if T is a bicontinuous linear isomorphism from E onto E such that Tx = x then
F 0 0 P(u T) = (Pu) o T for w i n l ;
(L ) P is x -efficient: i.e., v 3 0 (Pu)(xQ) = u(xQ) for u in X ; (L ) P is idempotent; i.e.
Pu = u for u in # .
Briefly stated, an ;r -value is a projection from J onto E which is
Global Theories of the Linearizing Projection 187 x -symmetric and # -efficient. If some topology is placed upon X so that E
becomes a closed subspace it may also be required that P be continuous. In this case P is called a continuous x -value.
2. The Polynomial Theory. A symmetric multilinear form of degree n on a real linear space E is a real valued function F on the n-fold product of E
such that (a) F is separately linear in each component, i.e.,
. , ax . + by . , x ., _ , . . . ) ,7 ,7 ' ,7 + 1 ' '
n ;
= aFxlS .
+ bF(xv.
a l l x , . .
* ' ' xj' x j + i ' ' '
· · , yü.,x; + l , . . .
. , x , y . in P:
. ,*„)
. , * )
a l l
P (x,
for all j = 1 , 2 , . . . , n ; all x , . . . , x , y . in E: all reals a and b and all j = 1 , 2 , . . . , n ; (b) the value of F does not depend upon the order of its components, i.e.,
[xl9 x2 , . . . 9xn) - \xx(i) ' xx(z) ' ' * * ' Xx(n)'
for all x , x2, . . . , x in P and all permutations x on (l,2,...,n). For x and y in E and Zc a positive integer less than n, F(x^-,yn~^) denotes the value of Fix. , . . . , x ) where for k components we have x . =x and for
1 n Λ η Λ J n-k components we have x . = y; P(x ) denotes P (x ,x , . . . ,x ) . The n-th
^ /s
degree homogeneous polynomial F derived from the multilinear form F is defined by the equation
F(x) = F(xn) . The multilinear form P is called the polar form of P. A polynomial functional H on E is a linear combination of homogeneous polynomials. The degree of a polynomial function is the highest degree of homogeneous polynomials in its expansion. Since each linear combination of homogeneous polynomials of degree n is either the zero polynomial or a homogeneous polynomial of degree n it follows that each polynomial H on E can be expressed in the form
n H(x) = ^ Fk(x) (2.1)
k=i where each F-, is either zero or a homogeneous polynomial of degree k.
A homogeneous polynomial P defined on a linear space E with norm II II is continuous if and only if there is M > 0 such that \F(x) | </kflUII for all x in P . A general polynomial H is continuous if each summand in its expansion is continuous. Let p(P) denote the linear space of all continuous real valued polynomials H on P . By our definition p(P) does not contain constant functions so #(0) =0 for every H in p(P) .
188 W. H. Ruckle THEOREM 2.1. Let x be a distinguished non-zero point in a Banach space E. There exists a unique xn-value P an p(E) given by the formulas
~ I / ^ \ I ds, for x in E , (2.2)
! \ (-) I . / U I 0
(PH)(x) = f [^^(sx0+ta:) t=o H in_ p(£);
PH)(x) = y F ^ U ^ " 1 ,a;) for x in E (2.3) fc-1
when H has the form ( 1 - 1 ) . The proof of this result may be found in (Ruckle [8]).
3. The Analytic Theory. An analytic theory of linearizing projections can be quickly obtained from the polynomial theory. For this it is necessary to introduce a topology and require that the ;r -value be continuous with respect to this topology.
A real-value function H on a real Banach space E is called real
analytic if there is a sequence (F ) of homogeneous polynomials with F
having degree n (or =0) and
Mn = sup \Fn(xl9. .. ,xn) \/(\\xl || . · .||a:J ); χ.Φθ for each «/<«>
(3 1) such that H(x) = ) Fn(x) (3,2)
Ίη
for all x in E . Because of (3.1) the series in (3.2) converges absolutely and uniformly on bounded subsets of E. Moreover, the sequence (F ) is uniquely determined by H. Let A(E) denote the space of all real analytic functions on E . A topology on A(E) is determined by the sequence of semi-norms .
Pk(H) = '] MnVn k= \,2,.. . (3.3)
where each M is given by (3.1). With this topology A(J£) is a complete metric linear space, i.e. an (F)-space which is, however, not a Banach space.
Let F denote the linear space of all homogeneous polynomials of degree n along with the zero polynomial considered as a degenerate homogeneous polynomial of degree n . Each F is then a closed subspace of A(E) .
The sequence (F ) forms a Schauder decomposition of A(E) in view of the unicity of the expansion (3.1). See (Ruckle [8].)
THEOREM 3.1. Let x~ be a distinguished nonzero vector in a Banach space E. There exists a unique x -value P on AE) which is continuous with respect to the topology determined by the seminorms (3.3). The x -value can
Global Theories of t h e Linearizing Projection
be given by the formula
(p*)(*) = l (^ ) f f K + t* ds t = o
or the formula (PB)\x) = V ϊ ( χ η - \ χ )
L ° n
189
(3.4)
(3.5)
for H given by (3.2).
PROOF. Existence. It will be shown that PH as determined by formula (3.4) is defined for all H in A(E) , satisfies Definition 1.1 and is continuous.
If H is determined by (3.2) then for all s and t in R
H[sxn+tx) = ) ) [ ] s t F (x : x ) . (3.6) n k=o
The series (3.6) converges absolutely and uniformly on bounded subsets of E
because the sequence (F ) satisfies (3.1). Therefore, we have
(£)«<
[(A)«
sxQ+ tx)
sx + tx) \ 0 J
L L ' ' n k=o
\k)Kn ^Xo ; * '
k=o z. n-i £, n-i N
J[fe)^V^ Jt = o /.., u (3.7)
All of the series encountered converge absolutely because the sequence (F ) satisfies (3.1). Therefore, PH exists for all H in A(E) .
It is clear that P is linear. Because of (3.7), PH satisfies (3.5) so that P must be x -efficient. If T is a bicontinuous linear isomorphism from E into E such that TxQ =xQ
P(HuT)x) = f [(A) HuT (sxQ+tx) t=0 ds
ι
- I [ ( Α ) , ( Τ Κ ) + ^ ) ) ]
[(^) *(**<>+tr*> t = o
t=0
P #) T^ .
T h e r e f o r e , P i s x - symmet r i c . Suppose | | icn | | < k ; then for each x i n E we have
190 W. H. Ruckle
\P(H)(X)\ < Y i ^ - 1 , * ) ! < £ «„ΙΙ^ΙΓ"1 11*11 n n
n which leads us to conclude that for all H in A(E)
II pcs)II <Ι*οΓχρ^) · This proves that P is continuous from A(E) onto E .
Uniqueness. By Theorem 2.1 an x -value is uniquely defined by the equation (3.4) for H in p(E). But p(E) is dense in A(E) so that if Q is any continuous xQ-value it must coincide with P on all of A(E) .
4. C —Theory. A real-valued function / on a Banach space E is said to be Frechet different table at x if there is a continuous linear functional x on E such that for each u in E x
x*(u) = lim \/t)f(x+tu)-f(x)). X t-+Q
For a general discussion of Frechet differentiability, see (Dieudonn£ [3]). The functi df (x;u) .
The functional x* will be written df(x; ·) and its x (u) its value at u,
1 Let C\E) denote the space of all real-valued functions f on E such that f (0) a 0 and the correspondence x-*df(x;·) is locally bounded and locally uniformly continuous from E into E the dual space of E, That the correspondence is locally bounded means that for each n = 1,2,..., we have sup \\df (x; · ) || : ||#|| < n <°°. That the correspondence is locally uniform continuous means that for each n and each 3>0 there is θ = Θ (n, 3) such that
\\dix;·) -dy;-)\\< 3 whenever \\x\\ and || y II are <n and ||χ —ζ/|| <θ. For example, ^ΛΕ) contains all real valued continuous polynomials on E. See Chapter 25 of (Hille and Phillips [4]). If E has finite dimension then local boundedness and local uniform continuity are implied by continuity since E is then locally compact.
The space CAE) can be given the topology of uniform convergence on bounded sets. This topology is determined by the sequence of seminorms
Pn(f)= sup||df(x;.)|| : \\x\\<n, n-1,2,...
This topology makes C (E) a complete metric linear space which is, however, not a Banach space. The Hausdorff property of this topology follows from the fact that if p (/) = 0 for each n then df(x;») = 0 for all x. Thus / is constant; but since /(0) = 0 it follows that f(x) = 0 for all x.
Global Theories of the Linearizing Projection 191 The following result is the main theorem of (Ruckle [8]). The proof
is lengthy so we omit it. The uniqueness portion of the proof depends upon 2
the fact that if / is a real valued function defined on R which has continuous k partial derivatives then the Bernstein polynomial for / along with its first k derivatives converges uniformly to / on compact sets. See (Kingsley [5]) or (Butzer [4]).
THEOREM 4.1. Let x be a distinguished nonzero point in a real Banach . !
space E, There is a unique continuous x - value P _ση CQ(E) given by the formula 1
(Pf)(u) = [ df(sx ;u) ds, u in E . (4.1) 0
5. Examples. This section contains three examples which illustrate potential applications of the theory described above. It is not claimed that the functions which appear realistically model the situations encountered. The functions are given in specific form so we can display the method of calculation. EXAMPLE 1: Risk. A certain polulation has four components. If a member of this population is exposed to x units of Hazard A and y units of Hazard B the expected number of days of life lost is
2 2 / x,y) = 3x y + 2xy for one from Component I;
f (x,y) = 4x y for one from Component II;
f (x,y) ~ 2x y + 3xy for one from Component III; 2 /, (%iy) = 2xy + x y for one from Component IV.
Each member of the population is exposed to 1 unit of Hazard A and 2 units of Hazard B and is to be compensated for expected life loss due to source A by Agency Alpha and life lost due to source B by Agency Beta. What is an equitable division of the compensation for the four segments of the population between Agencies Alpha and Beta ?
In this case E is R , the space X is the space of polynomials in two variables and the distinguished vector x is the point (1,2). By using the formula of Theorem 3.1 to compute Pf we obtain
1 0 —-1 3 (s + tx) ( 2s + iz/) d ~t /
+ 2(s+ tx)(2s+ty)
1 2 2 = J 20 s x + \\ s y ds 0
= (20/3) x + (11/3) y .
ds t = 0
192 W. H. Ruckle
S i m i l a r l y we o b t a i n
Pf2 = 3 ^ + 42/
Pf3 = ( 1 6 / 3 ) * + (11/3)1/
Pfk = (16/3)* + (4/3)2/ · When # = 1 and ι/=2 these formulae lead to the following results. Members of component
I receive compensation for 20/3 days from Agency Alpha and for 22/3 days from Agency Beta;,
II receive compensation for 8 days from Agency Alpha and 8 days from Agency Beta;
III receive compensation for 26/3 days from Agency Alpha and 22/3 days from Agency Beta;
IV receive compensation from 16/3 days from Agency Alpha and 8/3 days from Agency Beta.
EXAMPLE 2. Cost Allocation. Suppose the cost of generating electricity at the rate of x(t) KW/day for one day is given by the functional
1 2 C(x) = / x(t)) gt) at
0 where g is a fixed continuous function on [0,1]. If there are n users whose rates of use are x , x , . . . , x respectively, what is a reasonable
1 * 2 n (continuous) rate of charge for each user by a supplier who must recover his costs but not obtain a profit ?
Here E is the space C[0,1] the space of continuous functions on [0,1 ]; the space X is p : the distinguished vector x is given by
n x = ) x · ·
o L J We compute PC by the same procedure as in the previous example: P C (V = /[(^)P(^o + )]t = o <*«
1 r , . x 1
o u d H= I 3 \ 1 ί^τ) / (sxn (w) +tx .(u)) = g(u) du \ da \dtv o u d Ji=0 0
1 = f x (t) g(t)x.(t)dt.
0 U J Thus the rate of charge is given by the function #«(£) <?(£) which is intuitively reasonable.
EXAMPLE 3. Profit Allocation. A company has n departments. If the effort put forth by Department i over a given period is described by the continuous function u .(t) for i= 1 , 2, . . . , n then the company will realize a
Global Theories of the Linearizing Projection
profit given by the functional
193
1 sk(i) f(ui9u2,...,un) = J M l ( r ) " w . . . M n ( r ) f e ( w ) dr
where k(\) , . . . , k(n) are positive constants. What proportion of the profit should be attributed to each department ? Here the space E is the ft-fold product of C[0,\]; the space X is CQ (E) . To find Pf we apply Theorem 4.1; (the vector function (u ,,..,u ) will be denoted by u.
df(axiu) = [ ( ^ ) j TJisx^ + tu^r)^ dv 0 i
O) Hi) Hj)-d(i,t T T ( ^ , w +tu.(2·))Λ^-α^^β)1
J J J J t = ü J dr
Therefore,
here d(i,j) is the Kronecker delta which is 1 if i=j and 0 otherwise. Thus we have
1
df(sx;u)=\ £ M ^ r i ^ T T e ^ ^ ^ ^ i r b i J*
0 ^ «7
1
Pf(u) = [ dfsx\u) ds o
„ . 1 1 Y 7 / -N f / f (xk(j) )-i J i / N Ί—r / N^(J) -d(i9j) , = ^ k(t) I ( I ö is ! wx (r) j I x Ar) dy 'd dr i o o J
1
= I [*<*> T feÜ)] J «!<*·> T T x.vl·™-*^^ dr . ί Ί ° J'
The meaning of this is that the proportion of profit attributable to Department i is
194 W. H. Ruckle
REFERENCES
1. R.J. Aumann, and L.S. Shapley, Values of Eon-Atomic Games, Princeton, 1974.
2. P.L. Butzer, On two-dimensional Bernstein polynomials, Canadian J. Math, 5 (1953), 107-113.
3. J. Dieudonne, Elements dl'Analyse, Vol. 1, Ch. VIII, Gauthier Villars, Paris, 1971.
4. E. Hilie and R.S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. Colloq. Publ. 31 (1957).
5. C. Kings lev, Bernstein polynomials for functions of two variables of class CW , Proc. Amer. Math. Soc. 2 (1951), 543-554.
6. W.H. Ruckle, The infinite sum of closed subspaces of an F-space. Duke Math. J. 31 (1982), 543-554.
7. W.H. Ruckle, An extension of the Aumann-Shapley value concept to functions on arbitrary Banach spaces, Int. J. Game Theory 11 (1982), 105-116.
8. W.H. Ruckle, The lineraizing projection, global theories, IMA Preprint Series (1984),* 64.
9. L.S. Shapley, A value of n-person games, In Contributions to the Theory of games II, Princeton, 1953, 307-317.
ON ONE-PARAMETER GROUPS OF AUTOMORPHISMS OF VON NEUMANN
ALGEBRAS
A. B. THAHEEM Department of Mathematics, Quaid-i-Azam University,
Islamabad, Pakistan
1 . Introduction and Background. Let a, : t € R] and 3 : t € R] be one-parameter groups of ^-automorphisms of a von Neumann algebra M acting on a Hubert space H. Thaheem, Van Daele and Vanheeswijck [8] have shown that if a O)+0t_£(x) = $.f-(x) + $_-f-(x) f o r a 1 1 x^M ar d for all t€R, then a = 3, on Mp and a, = 3_+ on Af(l-p) for a central projection p in M. Such operator equations have also been studied for arbitrary pairs of automorphisms (cf. [6], [7], [10]). Some applications of these results are discussed by Haagerup and Skau [3] in the geometric interpretation of the Tomita-Takesaki theory and Haagerup and Hanche-Olsen [4] in the generalization of the Tomita-Takesaki theory for Jordan algebras. For an arbitrary pair of automorphisms, it has been shown in [7] (see also [10]) that if a and 3 are ^-automorphisms of a von Neumann algebra M satisfying α + α*"1 = 3+3""1 and a, 3 commute then a=3 on Mp and a = 3 _ 1 on M(\-p) for a central projection p in M. Apparently the result of [8] is more general in the sense that it does not use the commutativity of the automorphism groups. Paper [8] exclusively deals with this single theorem. The proof of this result depends on several lemmas based on the spectral subspaces theory of Arveson ([1], [11]). Recently the present author [9] has introduced a bounded linear mapping ψ associated to a one-parameter group of ^-automorphisms a, : t € R\
by the formula
ψω (cosh (TTt)) l at(x) dt , x£M).
In general the mapping ψ may play a role similar to the infinitesimal generator to study one-parameter groups of automorphisms, ψ is bounded but it has
195
196 A. B. Thaheem less algebraic properties as compared to the infinitesimal generator. We may remark that our mapping ψ is also similar to the analytic generator of one-parameter group introduced by Cioränescu and Zsido [2], It is shown in [9] that the mapping ψ reconstructs the group completely in the sense that if ψ (respectively ψ^) is a map associated to a, : t €R ] (respectively ßt:t €/?), then ψ = ψ implies that at + OL_t = β + (3_£ for all t£R.
This demonstrates the fact that the operator equation considered in [8] occurs naturally.
The purpose of this paper is to use the mapping ψ as in [9] to provide a simple and a short proof of the decomposition theorem of [8].
2. Results. Assume that α^ : t £ R and ββ : s £ i? are non-trivial one-parameter groups of --automorphisms of a von Neumann algebra M which satisfy the relation
H + α-t = es + ß_s · ··· (A)
Multiplying equation (A) by ßs first on the right hand side and then on the left hand side, we get that a, β +α , β = ß a +ß a , . Rearranging this
OS ~C S S o S o we get the following equation
a t ßS - ß e a t e sa_ t-a_ tß s. (1)
Equation (1) is one of the several possible consequences of the relation (A). But for our purpose here it is enough to consider (1).
To settle the notations and get the results, we recall a few definitions.
DEFINITION 2.1. ([9]). Let a, : t £R be one-parameter group of *-automor-phisms of a von Neumann algebra M acting on a Hubert space H. Define a mapping ψ on M by the formula
+ oo ΨΟ0 = J ^t
2 -7Tt at(x) dt , x£M) .
_oo e + e The integral is well-defined in the strong operator topology and gives a bounded linear operator on H. Considering any element y in the commutant U' of M, one can see that ψθτ) commutes with y and hence ψ(χ) € M. Therefore, ψ is a mapping from M into M. In fact, it can be shown that ψ is bounded (for details, see [9]).
(r) DEFINITION 2.2. ([9]). For any r > 0 , define an operator ψ on M by
_oo e + e \ a bounded operator of M into M ([9]).
To prove our main result, we need the following lemmas. As before ψ is a bounded operator of M into M ([9]
a
One-parameter Groups of Automorphisms 197
LEMMA 2 . 1 . For any r > 0 ,
(r) (r) _ (r) (r) = (r) (r"1) _ (,"1) (r) r a r 3 3 a 3 a r a ψ3
^ Ö Ö F . + 0 0 + 0 0 ^ (Ar) Ar)\ , , f f 2r 2r 0 , , , ,+ νψα Ψ 3 X)= *t -vt' πβ - π . at V * ds dt
N ' J J e + e e + e —OO —OO + 00+00 ^ s ^^_
N 7 -L irr, e + e e + e e + e e + e for a l l x in M.
Since the integrands are bounded Borel functions, therefore by Fubini 's theorem we get
2 is • KS ^etat*B-WMdedt
e + e (2)
From equations (1) and (2), we have + 00+00 ££ £s
Wp)*Btp)-*f)*aCr>fa- \ I ^ T ^ - ^ - i i i ^ ^ - a ^ ß ^ U ) ^ ^ 7 -oo -oo e + e e + e + 00 +co .
-it 0 ^s p 22^
n a, —a, 3 ) (#) ds dt. - o r, - oo β + < _π£ -ns -T\S s t t s
-oo -oo e +e e +e
This implies that
(*r *r-*r ·<> =(*r*r,,-*r'M->> for all x€M . This proves the lemma.
O) (r) LEMMA 2.2 a, and 3 commute if and only if ψ and ψβ commute. (r) O)
PROOF. If a, and 3 commute then ψ and ψ^ commute follows from equation (2) in the lemma above. To prove the converse, put
+ oo
^ ( t>=i vA -,t- J f - ^ \ h - h a t ^ x ^ ^ d s -oo e + e e + e
for any pair ξ,η in H. Then + oo Γ ritft) dt = 0.
— oo
Put v = e , then f(a) = 0 for all a. This implies that /=0 (see for instance [5]). Thus for all ξ and η, we get
198 A. B. Thaheem
2r ((atee-e8at)(ar)C,n)d8 = 0 I l\t ~l\t ITS -T\S - co β + β e + β
Applying Fourier transformation once again, we get
which proves the lemma. Setting r = 1 in Lemma 2.1 and s = t in Lemma 2.2 we get that the rela
tion a + a = 3 +3 , implies the commutativity of a and 3, for all t and V ~C t ~~t Ό u
hence by Theorem 2 in [10] (see also [7]) we get the main result of [8]. THEOREM 2.1 Let c^ : t O ) and 3^ : t € i? be one-parameter groups of ^-automorphisms of a von Neumann algebra M satisfying
for all t € R . Then there exists a central projection p in_ M such that af=$f £H MP §nd a = 3 , on M(l-p).
REFERENCES
1. W. Arveson, On groups of *-automorphisms of operator algebras, J. Funo. Analysis, 15 (1974), 217-243.
2. I. Cioranescu and L. Zsido, Analytic generator for one-parameter groups, Tohoku Math. J. 28 (1976), 327-362.
3. U. Haagerup and F. Skau, Geometric aspects of the Tomita-Takesaki theory II, Math. Scann. 48 (1981), 241-252.
4. U. Haagerup and Hanche-Olsen, Tomita-Takesaki theory for Jordan algebras, J. Operator Theory, 11 (1984), 343-364.
5. W. Rudin, Fourier Analysis on Groups, Interscience Publishers, New York, London.
6. A.B. Thaheam, Decomposition of a von Neumann algebra relative to a ^-automorphism, Proc. Edinburgh Math. Soo., 22 (1979), 9-10.
7. A.B. Thaheem, On a decomposition of a von Neumann algebra, Rend. Sem. Mat. Univ. Padova, 65 (1981), 1-7.
8. A.B. Thaheem, A. Van Daele and Vanheeswijck, A result on two one-parameter groups of --automorphisms, Math. Scand. 51 (1982), 261-274.
9. A.B. Thaheem, A bounded map associated to one-parameter group of ''«-automorphisms of a von Neumann algebra, Glasgow Math. J. 25 (1984), 135-140.
10. A.B. Thaheem and M. Awami, A short proof of a decomposition theorem of a von Neumann algebra, Proc. Amer. Math. Soc. 92(1984), 81-82.
11. A. Van Daele, Arveson's theory of spectral subspaces, Nieuw Arch. Wisk. 27 (1979), 215-237.
ON THE EULER-MACLAURIN SUMMATION FORMULA FOR ALTERNATING SERIES AND
A NEW PROOF OF THE FUNCTIONAL EQUATION FOR θ(χ)
P. L. WALKER Department of Mathematical Sciences, University of Petroleum
and Minerals, Dhahran, Saudi Arabia
1. Introduction. In the f i r s t part of th i s paper we review the Euler-Maclaurin summation formula in a form which is sui table for ser ies of real terms whose signs a l t e rna t e . The resu l t s are also extended to the case of two var iables .
In the second part we apply the formula to the the ta -ser ies
Q(x) = \ e L
— CO
when expressed as a doubly infinite product. To state the results we introduce the following notations:
N (i) For N = 1,2,3,..., let γ„ * Π n/n , here and throughout the paper n, is used as a convenient abbreviation for n - j ·
(ii) For m,n = \ , 2,3, ... , let
m2+n2 x2)(m\ + n\x2) %>nix) = , 2 , 2 2 W 2 + 2 2=%9m0,x)'
[m +n x )rn + n x )
The main result of the paper is that N N
Θ (x) = (\+x2)~llL>yN π Π qm nx) -> Q(x) a s N -> «> . m = 1 n = 1 '
As a corollary, the symmetry property of q (x) noted under (ii) shows that
x1/49ff(x) =χ-^ΘΛ7(1/χ)
and that consequently on letting #-»<», the same is true for Θ (x) . This
199
200 P. L. Walker gives a new proof of the well-known functional equation of Θ(x) : see [2], [3], [4] for alternative approaches.
Finally, we extend the calculations to give an explicit integral representation for Q(x)/Q (x).
2. The Euler-Maclaurin Formula for Alternating Series. We review a form of the formula which applies to series of the form
n 2(n-rn)
£ /W-/(r-i)= £ (-1)β/0π + 2β)· r=m+i s=i
(The choice of the scale factor x here makes the functions C defined below 2 n
have period 1 (instead of 2) and gives a closer analogy with Bernoulli polynomials .)
We begin by defining a function C, on [ 0 , 1 ) as follows: 1 - 1/2 , 0 < x < 1/2 , + 1/2 , 1/2 < x < 1 , 0 , x = 0, 1/2.
C is then extended to B by periodicity: C,(x) = C,(x - [x]).
Let / be a function with a continuous first derivative on [ 0 , 1] so that we can write
1 i 1 \ 0λχ) fx)dx =-\ I f\x)dx+\\ f'x)dx 0 0 1
= |/(0 -/(£) + (°)> 2' 1
/(!)-/(£) = jf(0-f(0)+ f C1(x)f(x)dx. 0
If j^ is continuous on [m,n] for integers m,n we can add the contribution over each interval [r , r + 1 ] to obtain
f /(r)-/(r-£)=if(n)-/(W)+j (*)/'(*)<** . 0) We note that the function C1(x) has the Fourier expansion on 1R given
by 2 ~ sin(2r+ Ι)2ττ*
2>=0 ^ 2 r + 1
We now integrate repeatedly and obtain a sequence (Cn(x)), n = 1,2,3, , of periodic functions on 1R as follows:
The Euler-Maclaurin Summat ion F o r m u l a 201
DEFINITION 2 . 1 . For x G I R , n = 1 , 2 , 3 , . . . , d e f i ne the f u n c t i o n s Cnx) by
_ 4 n i OD cos ( 2r + 1 ) 2πχ — η π / 2 C (a;) = V
(2π)η Δ ( 2 r + l ) n
o r 7, i C (a.) =
( " 1 4 ( 2 / Q ! ^ c o s ( 2 r + l ) 2 7 T x 2A: ( 2 π ) 2 Α : L 2r+\)2k
c ix) ( , ) * < 2 ? ; - l ) ! V s i n ( 2 r + 1 ) 2 „
v / p = 0 \ /
Then the f u n c t i o n s C x) have t he fo l lowing p r o p e r t i e s :
LEMMA 2 . 1 .
(i) When restricted to (0,-), C (x) is a polynomial of degree (rz-l), and C'(x)=nC , (x) there. (ii) For all real x, C (x) =B (x)-B (x - Ö) , where B are the Bernoulli functions ([l], Ch. 23). (iii) JFor 0 < x < | , C (x) = - n/2n En_1(2x) , where E is EulerTs polynomial ([1], Ch. 23).
PROOF, (i) is immediate by induction, and (ii) and (iii) easily follow by comparing the relevant Fourier expansions ([1], 23.1.16).
In particular we can define a sequence (C ) of constants by
DEFINITION 2.2. Let C = - \ , and for n > 2 , let
Cn = Cn(0) =Β (0)-Βη\)= 2Βη(\-2~η)9
We now take the formula
n /(*)-/(*-£) =\fn)-fm) + j Cl(x)f'(x)dx V
which is established above, and integrate the last term repeatedly by parts to obtain
THEOREM 2.1. Let / be k times continuously dif f erentiable on [m ,n] .
Then k
J /(r)-/(i-5)=i/(n)-f('n)+ J -i^f- C. [f U~ l \n) -f U~ l \m) + J=2
+ ~]\k*] ί Ckx)fWx)dx k k- m
j ! «7 I J k\
202 P. L. Walker
We now extend t h e s e r e s u l t s t o f u n c t i o n s of two v a r i a b l e s . Suppose
t h a t / i s a func t ion on 5 = [ 0 , 1 ] χ [ 0 , 1 ] whose p a r t i a l d e r i v a t i v e
f (u,v) i s con t inuous on S, Then
1 1 [ I CY(u) 0λ[ν) f u u , v ) d u d v
1
= ( ^ 1 Wi f 1 (« . l ) - f 1 (« . l )+ If^u^du 0
- | | / ( i . i ) - / ( ^ i ) + | f ( o , i ) - ^ ( i , | ) - / ( i , i ) + ^ (o
+ | | / ( i , o ) - f ( | , o ) + | / (o ,o )
0 0 1
which can be rearranged to give
/ ( i , i ) + f ( i , i ) - / ( | , i ) - f ( i , | )
= ψ ( ' . ι ) + f ( o , o ) - / ( o , i ) - / ( i , o )
J Cl(u)\fl(u,\)-f1(u,0)\du
1 J Cl(v)\f2(\,v)-f2(0,v))dv
+ ^ 0
+ Ί
0 1 1
r=m+l 8=n+i
+ J J c Au) cAv) fl2u9v) du dv. (2) 0 0
If we now suppose that / is defined on R = [m , mf ] X [ n , n1 ] then we obtain from (2) by addition over the subsquares in R the following result.
THEOREM 2.2. Let m ,m ',n, η' be integers and / be continuous on [m 9m '] X [η,η']. Then
1 s=n+i
+ i j ^(Μ ) | / (M,W')-/!(«,«) ^
j j ^(u) (u) f12(w,^) dw dv.
+ 2 n
+ m n
The Euler-Maclaurin Summation Formula 203
Ev iden t ly we can o b t a i n f u r t h e r terms i n t h i s expansion i f needed by
r epea t ed i n t e g r a t i o n by p a r t s as was done t o o b t a i n Theorem 2.1 from ( 1 ) .
3 . The T h e t a - S e r i e s Θ(x) . We de f ine CO
θ(χ) = V n= -co
for Re x > 0. For simplicity in the proofs we shall suppose that x is real; the proof can be easily extended to the complex case. In order to apply the results of Section 2, we write Q(x) in the form of a quotient of infinite products
QQ(x) Q2(x) Q(x) = , where V ; Gjte) Q3(x)
QAx) = Π ( 1 - e ) , Q(x)=J[(\+e ), 1 1
ρ (x) = Π (l+e ) , ρ ( χ ) - Π ( 1 - e ).
We can further expand each of these products to obtain
/ \ °° —?IT\X °° —ΥΙΊ\Χ °° / Ti X \ Q (x) = Π e · 2 s inh ηπα; = Π e 2ηιτχ Π I 1 + — ! ,
n= l n= l m= l ^
ρ (a:) = Π β ·2 cosh ηπχ = Π e · 2 n (1-1- j — ! , 1 n=\ n=\ m=\ ^ ml '
2 2 £ 2 ( ^ ) = Π £ * ' 2 c o s h n ι τ χ = Π e l · 2 n ( 1 + 2~ ) > a n d
7 2 / ΎΪ X \ i \ °° — Yl TTX °° — Yl-, T\X °° I 1 \
Qo W = Π e l ' 2 s i n h n ira? = n e 1 · 2n ττα; Π I 1 + γ- j · n=i n=l rc=l ^ '
Combining these results we have proved CO Yi CO
THEOREM 3.1 Qx) = Π — JI a (a;) , where n= 1 l m=l ( ^ 2 + η 2 χ 2 ) ( ^ 2 + η 2 χ 2 )
a (x) = ^m ,ΠΚ / 2 . 2 2 \ / 2 . 2 2 \
(m + n a: ) (m + n x ) as in the introduction. [Since the double product
CO . CO / v π η/η, π q \x)
n=i 77z= ι is not absolutely convergent, it is important to maintain the correct order for the products over m and n. ]
It follows from this result that we can write
204 P. L. Walker
-- Q \ χ ι #->*> n= i i rn= l
θ ( * ) = l i m Π - - Π #->*> W = 1 1 777 =
= Um \VN Π Π % nx)\ Π Π % Jx)\ (3)
w h e r e
n-\ 1 a s d e f i n e d i n t h e i n t r o d u c t i o n . We d e n o t e t h e s e c o n d o f t h e s e f a c t o r s by
AN(x).
DEFINITION 3 . 1 Fo r # = 1 , 2 , 3 , . . . , l e t
N N Ajx) = Π Π q (x) · Nx
W = l 77?=il/+l Then we have the following results.
THEOREM 3 . 2 TO
( i ) l o g ANx) = - l l o g ( l + * 2 ) - | ^ ( u )
N N 2
\\ cx(v) ^2v* Ίάυ + \ ολΜ\ ολ(ν)
u(u2 + N2 x2) du
2 2 ^ N + v x
( i i ) AN(x) -> (1 +χ2)~ι/^ a s #-»«>,
— h u v x 1
, 2 , 2 2 x 2 dv du ,
COROLLARY 3 . 1
( i ) e
1/4
r 2 N - l / 4
n = 1 777= 1 ' (a:) ajs_ #-*«> 9
( i i ) x θ ( χ ) = x
Proof of Theorem 3 . 2 . We b e g i n by w r i t i n g
iogAN(x)= y \ l o g (7 ( x )
n = l 777 = il/+l A7
y y j f ( m , n ) + / ( m 1 > « 1 ) - / ( m , ^ ) - f ( ^ , η )
n = l 777 = /l/+l
w h e r e / ( κ , ν ) = l o g ( l + ^ ) .
We now apply Theorem 2.2 to obtain N m ' r Λ
Y V |/(77?,n) +/(/r71,n1) -/(ττι,η^ -/(77715η)| n=l m = N+l
= £ /(*' .Ό +/U.0) -/(m' ,0) +/(/V,il/)
+ \ \ cxW)\f^W*n) -fl(u,o)^du
The Euler-Maclaurin Summat ion Formula N
\ J Cl(v)^f2(m'9v)-f2(Niv)^dv
205
o m' N
C1(u9v) fl2u9v) dv du, N 0
where C,(u,v) i s used as a convenient a b b r e v i a t i o n for C (u) C (v). We note
the v a l u e s of the d e r i v a t i v e s of / :
fx(u9v) 2 V X jy I \ 2VX r> I \ -kuvx2
2 2 2 u(u +v x )
2 ^ 2 2 U + V X
/ 2 ^ 2 2x2 (u + V X )
These are substituted in the above expression and we obtain on the right hand s ide
n72 2 i i o g ( 1 + ^ ) - i o g ( 1 + ^ ) + I clM-^Lf-
JQ ^lm/z + vzxi- N +v x J
m' N
^ J0 (u2+v2x2)
2 ) 2 dw
<f u if w . (4)
Evidently we can take a limit as m1 -*«> and 3,2(i) follows at once. 3.2 (ii) is deduced from this by showing that each of the integrals in 3.2(i) tends to zero as N->°° .
For the first two integrals we can use the elementary estimate a' a'
Cx(t) $t)dt< I j Φ'(*) dt , a a
which is immediate on integrating by parts and observing that
|C2(M)| <c2 = \
It follows that for example
f n ( \ Ν2χ2 J C \u) du
N u(u2+N2x2)
1 . /v x 4 N(N2 +N2x2)
since the integrand is a decreasing function of u (recall that x is real and positive).
Similarly N
i N + u x < - · 2 Max
V N2+V2x2 4 *
To deal with the double integral we integrate by parts twice to obtain
206
where
a1 b'
\ \ a b
P. L. Walke r
a' b'
\ Cx[u ,v) <\>(u ,v) du dv = \ du ,ν) Φ 1 2 (^ ,v) du dv ,
,. a b x y
d(x,y) = ] I C1(u,v) du dv = ^ x y , 0 0
and x == inf | x-n | . (Note t h a t C2(x) = q·—x so d(x,y) i s not C2(x) C2(y) .
Hence n£ZZ a'bf a1 b'
\ C^u >v) §u ,v) du dv\ < yg- | Ul2u,v)
a b a b
and so the integral in (4) is dominated by co N
du dv ,
Τξ\ l \fU22^U*V^\dvdu
However
\fU22(u,v)
N 0
\2x2uh -t>u2v2x2+vh xh) / 2 _, 2 2x4 (u + > a; )
36 x (u2+?;2x2)2
and we obtain the estimate o o DJ
J l N 0
, 2 _,_ 2 2 , 2
oo oo , ^ 9 f ( [ X dV lu < Ϊ U , 2 2 :
^ x 0 (w + tf a: CO f TT a: j 9 π χ η
-r · —*- du = «- -*- 0 J 4 w3 32 N2
N
du
a s i\/->oo .
This completes the proof of Theorem 3.2(ii). Corollary 3.1(i) follows at once from 3.2(ii) and equation (3), and
3.1(ii) follows since .1/4 N N
1A / \l/4 ίν lv
i s unchanged when x i s r e p l a c e d by 1 lx.
We f i n i s h t h e s e c a l c u l a t i o n s by prov ing
THEOREM 3 . 3 . £ o r /!/= 1 , 2 , 3 , . . . , 717
C Au) log \ n , x = — aw - C. (u) — + ) u du
2 2 9 2 _,_ 2 / x N x + u
— kuvx , 2 ^ 2 2 x 2 (u + y x )
du di? ,
where 5 = [u , f ) : u > N or u > t f ]
The Euler-Maclaurin Summation Formula 207 PROOF. We begin by r e f e r r i n g to Theorem 3.1 which we can w r i t e in the form
r N N <> r N CO .
ex) = \ΎΆ7 Π Π q (χ)\ \ Π Π q (χ)\ η=ι m=i J ^η-ι m = N+i J
X ^ Π Π q (a;) Π ^- II q χ)\ . (5)
The first of these products is (1 + x ) Θ (#) , the second is A (x) , the third (by the symmetry of a (x)) is A„(\/x) 9 while the fourth we shall denote by B (x) :
CO CO BJx) Π — Π q x) ·
The aim of the proof is to develop an integral representation for B (x) as AT we did previously for A?(x) in 3.2(i). We begin by taking logs:
logzy*)
with
. \ " L n = N+l
CO
W = /l/+l
/(*,*) = log (l+^-f-) as before. The term log^— makes us follow a slightly different route than in the proof of 3.2. ,
m T h e S U m V fm,n)-fmi,n))
m = N+l
is equal by equation (1) (the case k=\ of Theorem 2.1) to m'
\f(rn' ,n) -fN9n)\+ j C (u) f^u ,n) du , N
and we may l e t m1 -*<*> to o b t a i n CO CO
£ / ( r o , n ) - / ( τ τ ^ , η ) = - £ / ( # , n ) + [ ^ ( u ) ^ ( u , n ) d u .
w = /l/+l N
We put this into the above expression for log B Λχ) and obtain ΛΓ
log BN(x) = lim n y -»Co
£ [log --| /(»,«) -fiff.nj) CO
+ j Cl(u)fi(uin)-fi(u , η ^ Ι ^ Ι .
η = Λ/+1
This sum over n can be written
208 P. L. Walker
m=N+l
Y l o g ( i - ) - i y f(N,n)-fN,ni)
+ [ Cx(u) ^ flu9n)-fl[u9nl)]fdu9
and we now apply e q u a t i o n (1) to each of t h e s e terms to o b t a i n
/ n CAU) 1 n n' l
du j " n
fN9n') -f(N,N)+ j C^v) f2(N,v)dv)i
oo
J Cxu) \\ ^ ( « , η ' ) -fl u9N) + J Cl(v)fl2u9v)dv\du.
(6)
Now as n' -»<» 9 the terms
I , . . « _ _ i / ( f f , M / ) = i l o g « _ _ i log N ** log 1 + 2 - a F - 5 - 8 V T — — j
n / 2 x 2 \
1 tend to — - log x . Also the terms
j —JJ— du + | I ^ W ^ l ^ n ' <iw
can be written
f C^u) n'2x2 1 u(uz +n'z χΔ J
dw ~ C, (w)
— ^u J . u
oo
j Cju) u du N u2 + nf 2 x2 't
r CY(u) du ,
J u and the first of these integrals is bounded by
2 Max -^ 0 as n' -> oo. ^ ^ « ) = __L w V 2 + n / 2 x 2 / In',
Thus we can l e t n ' - x » in (6) t o o b t a i n oo
log BN(x) = - ^ l o g x + I l o g ( l +x2)-\ | ^ ( v ) / 2 ( t f , t > ) dv N
^ ·Ν Ν
(The double integral is convergent independently of how the limits are taken, as was seen in the proof of 3.2.)
This gives us the desired expression for B (x) in the form
The Euler-Maclaurin Summat ion Formula 209
o 5° 2 V 2 "
•L t . 717 + T) Ύ* *7 / l / 2 + z ; 2 x 2 « ! u(u2+N2x2) du
N IM ^v x N QO oo o ( „ / ,\ — buvx
C I ( M ' ^ , 2 2 2 J J (w + f x ) tf N
We now w r i t e (5) in the form
1 ο β θ ( * ) = l o g ( ( l +x2)lheNx))+ l o g ^ ( x ) + l o g ^ ^ ) + l o g B ^ a : ) ,
and s u b s t i t u t e for l o g ^ , , log B from ( 3 . 2 ( i ) and ( 7 ) . This r e s u l t s i n
l 0 g r e i = - i °1U) ΪΧ\ 2 dU~\ C1V) 2VXl 2 dv ( f r o m ANM)
V ^ >N l u(u2 + NZ x2) J 1 Nl+vxZ N
- [ CM Nl du-\ C^v) — - | _ d „ (from A (1))
- J Μ*> -ΓΓττάυ + ί ^ w —ft2"2;dM (from v*» N N +v x N u(u + N x )
+ f [ C\ (u , i;) ^ ^ dw Jz; V X (u 2
+ t ; 2 * 2 ) 2
(on combining the three integrals). These terms combine (or cancel) in pairs to give
,72 logFliT = " ^ ( u ) — 2 — 2 ~ Y " + ~ — — \ d u
®NW N
l lu(N + x u ) N2+x2u2j
N r N N
^N2 +v2 x2 N2 x2 +v2^
+ J J C^u.v) 2
—huvx , , du dv , , , , 2 2 2N2
S (u + v x ) N
The first of these integrals is simply
and 3.3 follows. N The author is pleased to acknowledge the support given to this research
by the facilities at the University of Petroleum and Minerals.
210 P. L. Walker
REFERENCES
1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington D.C., 1965.
2. T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976.
3. C.L. Seigel, A simple proof of η (-1/τ) = (τ/i)1 η(τ), Mathematika 1 (1954), 4.
4. P.L. Walker, On the functional equations satisfied by modular functions, Mathematika 25 (1978), 185-190.
A FINITE ELEMENT APPROACH TO NONLINEAR TRANSIENT PROBLEMS
A. A. AL-KHARAFI* and D. J. EVANS Department of Computer Studies, Loughborough University of
Technology, Loughborough, Leicestershire, UK
Abstract. In this paper, the finite element method using six node quadratic isoparametric elements is used to solve a selection of nonlinear hyperbolic and parabolic equations which represent various special cases of the general nonlinear convection-diffusion equation. Those discussed are: (1) a quasi--linear convection dominated problem, (ii) a quasi-linear diffusion-reaction problem, (iii) a quasi-linear advection-diffusion problem (Burgers1 equation). In addition to these cases, linear analogues are presented and solved to emphasize the effect of the nonlinearity on the complexity of the solution. The features of the general model and the finite element technique used to solve the problems are discussed.
The above equations are solved in one or two space dimensions as the solution techniques are similar. Discussions and results are presented, and it is confirmed that the method is competitive relative to finite difference and other finite element methods.
1. Introduction. Initial/boundary value problems, which play a very important role in Applied Mathematics and Mathematical Physics, model most of the world dynamics problems. In many areas of Applied Mathematics the quantities of interest are often to be found as the solution of certain initial/ boundary value problems. Although, the analytical solution for a variety of simple initial boundary value problems which consist of one or many partial differential equations subject to initial and/or boundary conditions can be obtained, it was noticed (Zienkiewicz, o.e. [24]) that the actual solution for transient problems was generally difficult and indeed not available for nonlinear situations. Apparently, this availability depends on how severe
Present Address: Department of Applied Sciences, Faculty of Technological Studies, Kuwait.
211
212 A. A. Al-Kharafi and D. J. Evans the nonlinearity is. The analytical solution to the problems considered in this paper is presented to serve in assessing the solution accuracy. The finite element method is one of the recent numerical tools to tackle such problems in which the solution is approximated at a finite number of points and then interpolated to find the quantities of interest in the given region. In [1], we have used this method to solve a nonlinear elliptic problem incorporating the Davidenko-path procedure without which convergence could not be obtained in a region where the nonlinearity is very pronounced indeed. Yet, parabolic and hyperbolic equations might be considered as different extensions of elliptic equations, namely by adding different terms to the elliptic equation. In a computational sense this is to save the factorisations and calculations completed for the elliptic partial differential equation associated with the parabolic or hyperbolic partial differential equation and account for the extra time dependent terms. Accordingly, the structure of the resulting system of equations propagate forward the solution of the problem from one step to the next in a step-by-step fashion.
In this paper we will consider a selection of hyperbolic and parabolic nonlinear equations which represent different special cases of the general nonlinear diffusion equation:
-|£+ u-Vu = V(£Vu) + S , (1.1) σ V and/or some analogous extensions to these cases, where u is a function of
the spatial variable x(9y) and time t , D is the diffusivity and S is the source term.
An individual section of this paper will be devoted to considering each case of the above-mentioned cases of interest namely, the (i) Quasi-linear convection dominated problem
|2£+ u.yu = 0 , (1.2) a V
(ii) Quasi-linear diffusion problem, (diffusion-reaction equation),
| ^ = V-(Z)Vu) + fx*t9u) , (1.3)
(iii) Quasi-linear advection-diffusion problem (Burgers' equation),
|^4 u^u =4~ V2u . (1.4) dt Re Corresponding to the solution of these cases, linear analogue equations are
presented and solved to serve in highlighting the effect of the nonlinearity on the complexity of the solution.
These six equations are solved in one or two space dimensions as the solution technique is similar for both cases. Discussions and numerical
A Finite Element Approach to Nonlinear Transient Problems 213 results concerning the nonlinear problems are presented in Sections 4,5 and 6.
2. The Features of the General Model. The main features of the general nonlinear convection-diffusion (Burgers') equation (1.4) may be summarized as follows: (i) It can be considered as the core of the five consecutive cases governing a large number of physical, applied mathematical and engineering problems. These five problems differ gradually by the order of the spatial derivatives appearing in the governing equations, namely the
1. Convection dominated diffusion equation,
Diffusion equation,
ut +uux =0 , (2 .1 )
u , (2 .2 ) XX
3. Convection-diffusion equation (Burgers' equation),
u, + MM = yu , (2.3) t x xx 4. The equation describing the long time evolution of small finite
amplitude dispersive wave (Korteweg-de Vries) equation, u,+uu + u = 0 , (2.4)
t x xxx 5. Approximate equation for long nonlinear waves in a viscous fluid,
u.+uu +yu 4- 3w + 6u = 0 , (2.5) u t-L Jb %L· %L· %L· JL %L· tL· JL JL
derived by Topper and Kawahara [21], where γ, 3 and 6 are parameters.
It is clear that a wide variety of problems are governed by special cases of eqn. (2.3) (e.g. eqns,(2.1), (2.2)) or generalizations of eqn. (2.3) (e.g. eqns.(2.4), (2.5)). (ii) It is a good model for testing several numerical techniques since it
is one of the simplest nonlinear PDE's of which the exact solution is available and simply derived for combinations of a wide range of initial/boundary conditions. (iii) It is a good model that comprises the representation of elliptic problems (e.g. steady state convection-diffusion equation), parabolic problems (e.g. diffusion equation) and hyperbolic problems (e.g. convection-dominated diffusion equation). This behaviour depends on the magnitude of the various terms in the mathematical model. (iv) It admits the Hopf-Cole transformation [13, 9] given by
"=-έτ· ( 2 · 6 ) where Θ is any solution of the diffusion (heat) equation
214 A. A. Al-Kharafi and D. J. Evans
*± = ±£±. (27) 9* Re , 2 K
ÖX
This feature of Burgers' equation plays an important role in the determination of its exact solution when it is too difficult to be obtained.
These features have attracted a large number of authors to investigate the model and its special and general cases. Some of this work is referenced in this paper and used, along with the exact solutions, to show that the method described in this paper is competitive relative to other numerical techniques (e.g. the finite difference method or other finite element methods). 3. The Finite Element Technique. The core of the program used to solve the parabolic problem may be considered as an extension to the one used to solve the associated elliptic problem when the term including the rate of change of the dependent variable is ignored. So, initially the basic finite element techniques for elliptic problems is described.
The solution of an elliptic problem with a dependent variable u and independent variables x,y in a 2-D arbitrary region R of which dR, and dR2 are two distinct portions of the boundary Γ, is equivalent to the minimization of the functional,
I[u) = I I Ιλ (ux,u 9u) dx dy + I I2(u) du (3.1) R dR2
with 2
u = u on dR^ . (3.2) The region R is automatically triangulated and refined using six-node
quadratic, ten-node cubic or fifteen-node quartic triangular isoparametric elements. A set of basis functions (Ψ , . . . , Ψ ) is constructed such that Ψ is zero at all nodes except node number J. The solution u is expanded in a series of basis functions and substituted into the functional (3.1). By equating the gradient of I(u) to zero, results in a system of mN nonlinear equations for the mN unknown coefficients in the expansion of u9 where m
is the number of coefficients in the class of polynomials over which the minimized integral is taken. Newton's method is used to solve this nonlinear system, while the damped Newton's method is also used for those nonlinear problems of which the convergence of Newton's method is strongly dependent on the choice of the initial guess solution.
As mentioned previously, for the consideration of parabolic or hyperbolic problems, the basic technique is to account for the extra terms which are time dependent. In this case, we end up with a system of ordinary differential equations of which the unknown coefficients are now functions of time. The implicit Crank-Nicolson method is used to discretize time, and a
A Finite Element Approach to Nonlinear Transient Problems 215 Richardson extrapolation procedure may also be implemented to increase the order of convergence. In either case, the system of implicit, nonlinear equations must be solved at each time step using one iteration of Newton's method [19].
The solution technique commences by the triangulation of the region R
in two dimensions (or one dimension). The former case is obvious, but the latter requires an assumption which allows the one dimensional problem (in the x direction, say) to be triangulated, that is to assume another dimension (y, say) of height yQ so that the rectangular region thus produced (x <x <x2 , 0<y<yQ) is triangulated (Fig. 3.1) with all functions independent of y and the boundary conditions at z/ = 0, yQ determined from the fact that the normal derivatives of all unknowns will be zero there.
(Ο,Ο) χ 2
FIG. 3.1 Triangulation of one-D region R as a rectangle
Applications. In the three following sections applications of the F.E.M. will be made to many fluid dynamics problems using six-node quadratic triangular isoparametric elements. As shown in [1] they are the best choice amongst the quadratic, cubic and quartic elements that produce a reasonably accurate result in a short time. The region in which each problem is to be solved was chosen in such a way that allows the changes in the considered phenomenon to be visualized so that the performance of the F.E.M. may be sufficiently assessed. Two examples of this choice are (1) the consideration of the region in which the problem possesses a discontinuity at one end and tends to the steady state in the other (Section 5), and (2) the consideration of another region in which the solution changes from a steep to a flat front as shown in figures corresponding to each problem. The latter case stems from the fact that the conservation laws for mass, momentum and energy lead to both convective and diffusive terms in the resulting governing equations and the differing nature of these terms is the primary source of the difficulties encountered in the numerical solution (Section 6). A similar approach was
216 A. A. Al-Kharafi and D. J. Evans applied to choose the values of time at which the solution is presented and the time steps were chosen to produce sufficiently accurate results.
4. A Quasi-Linear Convection Dominated Diffusion Equation. In fact, the hyperbolic convection dominated diffusion equation
du o U n / / i \ — + u ^— = 0 , (4.1) at dx
may be considered as the inviscid version of Burgers' equation (1.4) and models the convection of disturbances in inviscid flows and is called the momentum equation in gas dynamics, where the effect of viscosity is usually neglected and possesses the so-called "transpositive property" which restricts any advected perturbation to be in the flow direction.
Consider the convection dominated-diffusion equation !τ + " ^ = ° > 0<χ<\, 0 < t < l , (4.2) o t dx
ux,0) = -x , 0 < x < 1 , (4.3)
w(0,t) = 1 , 0 < t < 1 . (4.4)
The a n a l y t i c a l s o l u t i o n of eqn. (4 .2 ) i s given by :
u(x,t) = 1 , x < t , (4 .5 )
u(x9t) = ^ T > t < x < \ . (4 .6 )
The F.E.M. presented in Section 3 was applied and the numerical solution u
was plotted (Fig. 4.1) versus x at typical values of the time t, compared with the exact solution, where the time step was taken to be At = 0.0125. These results agree closely with those presented in [8] of which similar relations were plotted using three different numerical methods, namely, a semi--implicit method based on a Lagrangian point of view and applied on an Eulerian grid system, an upwind method and Leith's method, while another finite difference "leapfrog" approximation was used by both Ames [2] and Richtmyer and Morton [17] to solve equation (4.1). This solution shows an exponentially-like divergence property in several cases.
A linear analogue of equation (4.1) governs the disturbance travelling with speed o and can be written as follows;
| i f f l | i = o , (4.7) ot dx
where the exact solution of (4.7) represents the lines on which its characteristics are constants and it may be written as,
φ = x - ot . (4.8)
The initial and boundary conditions are corresponding to the exact solution.
A Finite Element Approach to Nonlinear Transient Problems
FIG. 4 . 1 F.E.M. s o l u t i o n ( - · - · - ) , exac t ( ) , Δ£ = 0.0125
0 1 2 3 4 5 6 7 8 9 10 * X1CT1
FIG. 4 .2 F.E.M. s o l u t i o n of eqn. ( 4 . 7 ) a t £ = 0 . 1 , 0 . 6 , wi th c = 1.0, At = 0 . 1
218 A. A. Al-Kharafi and D. J. Evans The F.E.M. solution of eqn. (4.7) is plotted (Fig. 4.2) at typical values of time and is found to coincide with the exact solution. A comparison between the solutions of eqns.(4.2) and (4.7) is shown in Table 4.1 and appear to emphasize the effect of the nonlinearity on the complexity of the solution.
Table 4.1'- Comparison of the F.E.M. Solutions of Nonlinear and Hyperbolic Equations
Equation
4.2 4.7
Computation time*
31 sec. 31 sec.
l2 (error)
t = 0.2
0.13832 X10~2
0.0 **
t = 0.4
0.31713 X10-2
0.0
t = 0.6
0.54748 X10"2
0.0
For the values presented in this table, the solution is computed for ten steps with At = . 1 .
&& Zero error indicates that l2 < ε , where ε is the machine epsilon.
5. Nonlinear Diffusion-Reaction Problem. In the last few decades, attention has been turned towards the applications of the diffusion-reaction equation, as it models a wide variety of industrial problems in chemical, mechanical and biological processes. Meanwhile, great developments have been made depending on the use of numerical techniques in order to approximate the solution of the mathematical models governing such processes. The material presented in this section includes two examples of how those numerical techniques were widely facilitated by the use of digital computers.
An application of the diffusion-reaction equation is the accurate calculation of the flame speed which is necessary for determining the fuel consumption rate in a chemical reactor. The analytical solution of the model is rarely obtained because of the nonlinearity of the governing equation, so we present a simple model of which the exact solution is available in order to use it in the assessment of the accuracy of the numerical solution.
Travelling front solution. Consider the problem
du d2u , , N -r = — - T + u( \-u) , 9 t dx2
U(~ca,t) = 1 ,
u(+°°9t) = 0.
t > 0 ,
The exact solution is given by
where, uz) = ( 1 + A e )
x-ot .
A > 0 ,
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
A Finite Element Approach to Nonlinear Transient Problems 219
So u(x,0) = (1 +Aex//1)-2 , (5.6)
c i s a parameter r e p r e s e n t i n g the t r a v e l l i n g wave speed and chosen in t h i s
case to be 5/v6 by Ablowitz and Z e p e t a l l a (1979) .
The t r a v e l l i n g wave i s p l o t t e d ( F i g . 5 .1) a t t y p i c a l v a l u e s of t ime t
with 4 = 1 . 0 .
xicr1
Π I I I I I I I | I I I I I I M 1 |
2 3 4 X 1 0 1
FIG. 5.1 Travelling wave at typical values of time t, with 4 = 1.0, c = 5//s
Table 5.1 includes the root mean square (RMS) error of the F.E.M. solution given by
71 fu„-u,)2,l RMS \
L (5.7)
where u . is the numerical solution and u is the corresponding exact solu-tion and N is the number of entries in the error vector.
Table 5.1: Error in F.E.M. Solution of Eqn. (5.1)
RMS ( e r r o r )
t = 0.1
0.20427X 10"1
t = 0 .5
0.92924 X 1Ό-1
t = 1.2
0.18662
220 A. A. Al-Kharafi a n d D. J. E v a n s
6. Quasi-Linear Advection-Diffusion Equation. The so-called Burgers' equation (1.4)
du du_ 1_ d U_
was named after J. Burgens who thoroughly investigated this equation and related systems of equations in a remarkable series of papers during the period (1939-1965). Although, equation (1.4), where u = u(x,t) first appeared in [3], the complete solution of Burgers1 equation became known in 1950 [4] and since that time it has been a focus of attention of many investigators and some of their work is summarised in Table 6.1. The presence of the term \/Re9
where Re is the Reynolds number, in equation (1.4) is the source of many important characteristics of turbulence connected with the balance of energy and the appearance of the dissipation layers. Since these characteristics control the behaviour of the solution, then the parameter Re plays an important role in determining the step front solution and this feature is a primary reason of considering the Burgers' equation for different values of Re, In many related literature, the term 1 /Re is replaced by the viscositylike parameter V.
Table 6.1: Some Numerical Methods used to solve Burgers' Equation
INVESTIGATOR
Caldwell, J.
Caldwell, J. and Wanless, P. Caldwell, J. et at.
Donea, J. et al.
Evans, D.J. and Abdullah, A.R. Fletcher, C.A.J.
Lohar, B.L. and Jain, P.C. Nguyen, H. and Reynen, J. Spehrnoori, K. et al.
Varoglu, E. and Finn, W.D.L.
REF.
[5]
[6] [7] [10]
[11]
[12]*
[14]
[15]
[18]
[22]
METHOD USED
Explicit F.D.M., implicit F.D.M., fixed nodes F.E.M. and moving nodes F.E.M. Fourier series approach incorporating the method of lines. Variable mesh F.E.M. Taylor-Galerkin method.
Group Explicit F.D.M.
Traditional Galerkin method, spectral method and Galerkin F.E.M.
Variable mesh cubic spline technique.
Least-square weak formulation method.
Lumped and unlumped F.E.M. incorporating the system integrators. Weighted residual F.E.M. incorporating the method of characteristics.
k A broad discussion of Burgers ' equation is presented in this paper.
A Finite Element Approach to Nonlinear Transient Problems
Consider the e q u a t i o n ,
du , du \ d U n ^ ^ i dt ox Re ΰχζ
subject to the condition u(09t) = 0 ,
with the initial condition at time, t= 1 given by
u(x,1 ) = — 1 +exp [^(/-l/4)
The exact solution to Problem 1 is given by
x/t u(x,t) =
221
(6.1)
(6.2)
(6.3)
(6.4)
Numerical and analytical results at typical values of the time t and Reynolds number Re using NE number of elements are compared graphically in Figs (6.1)— (6.4)»where the F.E.M. solution is plotted using a dashed-dotted line and the exact solution is plotted using a solid line.
X10"
FIG. 6.1 F.E.M. (-.-.-), exact (- -) solution, Re = 500, At =0.025, NE = 160
222 X10 - 2 A. A. Al-Kharafi and D. J. Evans
SOj ' ' ' I i . i . I i i . I . i . I i . i . I i i . I i i i . I i i i . l t .» . I i .
FIG. 6.2 F.E.M. (-·-·-), exact ( ) solution, He = 200, At = 0.025, NE = 160
I i i i i I i < i i I i i i i I i i i i I i i i i I i t I i 1 i i i i l.i i i i
FIG. 6.3 F.E.M. (-·-·-), exact ( ) solution, Re = 750, At = 0.025, NE = 160
A Finite Element Approach to Nonlinear Transient Problems 223
O4s; x 4 0 :
35i
30:
25:
u 20i
15:
10:
5:
0 :
C
r
/ /x-b\
pi i i i I I i i i | I » i f | i i i T | i i i i f "
I 1 2 3 4 5 1 1 1 1 ( 1 1 1 1
6 ;
I
L··^ r i \1
1 1
r i m
\ F
i I
3 9 1 0 x X10"1
FIG. 6.4 F.E.M. ( ), exact ( ) solution, Re = 1000, At =0.025, NE = 80
The linear analogue of Burgers' equation
|£+M^=DÜ|, (6.5) 9t dx 3^2
models the solute dispersion in a uniform flow field in a homogeneous medium. The flow is made one dimensional by imposing the no-transport boundary condition. Equation (6.5) is solved subject to the initial condition,
cx,0) = 0 , (6.6) and the boundary conditions
o(0,t) =o0 , t> 0 , (6.7) c°°>t) = 0 > £ > 0 , (6.8)
where c is the solute concentration and D is the longitudinal dispersion coefficient and u is the flow velocity in the x direction. The problem is solved in the region 0<x<100, 0<£ and cQ9 D and u were taken to be of values 10, 1, .1, respectively.
The analytical solution for eqns. (6.5)—(6.8) has been derived by Ogata and Banks [16] and is given by
/ , \ "0 J I ux\ . (x + ut\ , r (x c(x9t) = — i exp [ -^ j erf c j = — j + erf c j — J lux 0 1 ^ ΛΡ \ τΓ j c i i i ; i ι τ e r i c i
where erfc(s) is the complementary error function and given by
= ) )
erfc (2) = — f ß""2 /π J 6?W .
(6.9)
(6.10)
The solute dispersion is solved for at typical values of time using eighty
224 A. A. Al-Kharafi and D. J. Evans quadratic triangular elements and plotted in Figs. 6.5, 6.6 compared with the analytical solution and another finite element solution to the problem which is described in [23] and based on the use of ten linear rectangular elements.
A comparison between the solutions of eqns.(6.1) and (6.5) is shown in Table 6.3 in terms of the CPU time of computation and the Euclidean norm of the error vector.
Table 6.3 Comparison of F.E.M. Solutions of Nonlinear and linear parabolic equations
Equat ion
6.1
6 .5
Computation t ime*
1 m i n . , 02 s e c .
1 m i n . , 0 .2 s e c .
l2 ( e r r o r )
t = 1 .0
0.20125
t = 10
0.43639X 10"1
t = 3 .0
0.16343
t = 30
0.69671X 10- 1
i = 5.0
0.07341
t = 50
0.54854X 10-1
For the values in this table, the solution is computed for ten steps with At = 1 .0.
7. Discussion. The effect of the presence of the non-linearity on the solution appears in the results presented in Tables 4.1 and 6.3. Although, the CPU time of computation is the same for an identical number of time steps to solve both linear and nonlinear partial differential equations, the accuracy of their numerical solution is clearly different as in the former case the results are of better agreement with the exact solution. This is to be expected since the solution of the nonlinear problem is obtained to an accuracy of At, the time step.
The integration of the solution within the domain of interest is indeed a good criterion of the accuracy of the solution as it indicates the status of the dissipation of the system or the conservation of energy in some cases and the flow rate in others. The behaviour of both of which can be anticipated from the theory. Another integration may be used to assess the accuracy of the solution in diffusion-reaction problems where the integration of the source term along the domain of interest may be related to the travelling wave speed.
The solution of Burgers' equation possesses a travelling discontinuity that shifts the centre of the shock wave in the flow direction and may cause oscillations. There are many strategies to cope with this difficulty and some of which are: (i) The classical approach of using a mesh sufficiently refined to cope
A Finite Element Approach to Nonlinear Transient Problems 225 I i i i i I i i i i I i i
FIG. 6.5 F.E.M. ( ), exact ( ) , F.E.M. [23] ( ) solutions, cQ = 10,D=l,u = 0.1tät = 1.0
(Note: (-·-·-) and ( ) are coincident)
10 15 20 25 30 35 40 45 50
FIG. 6.6 F.E.M. ( · - ) , exact ( ) , F.E.M. [23] k4AA D ( ) so lu t i ons , cn = 10, £> = 1, u = 0 . 1 , At = 1.0 MAA—P U
226 A. A. Al-Kharafi and D. J. Evans with the discontinuity [5, 10, 11, 12, 15 and 22].
(ii) The use of a variable mesh that allows the elements to have various sizes as the location of the discontinuity moves along the x-axis. This can be applied for various numerical techniques as F.E.M. [7] and cubic splines [14].
(iii) The use of unlumped finite element schemes as it is superior in controlling the oscillations and in representing the fronts [18].
(iv) The use of an upwinding approach in which some terms are added to (or multiplied by) the interpolation functions ((F.E.M.) or the difference molecule (F.D.M.)) in order to suppress oscillations arising in the numerical solution. In this paper we have used a fixed-node mesh with uniform triangulation
for smooth problems and nonuniform triangulation for problems possessing a discontinuity. In the latter case, the mesh refinement is adjusted to be dense around the discontinuity and uniform elsewhere in the region and the number of elements used in the triangulation varies, accordingly, from 40 to 360. This triangulation was sufficient to suppress oscillations arising in the convection-diffusion processes for large Reynolds number.
Many comments on the solutions of BurgersT equation may be made. These are as follows. The results presented in Section 6 visualize the property of Burgers' equation which states the balance between the convective and the diffusive terms. The solution may start with a steep discontinuity but it flattens out as time increases by the effect of the diffusive term and on the other hand the solution may also start with a smooth initial condition but the solution fronts become steep as time increases and tends to be asymptotic to x = 1 in the steady state.
The Reynolds number is the leading parameter in determining the behaviour of the solution and in many cases the solution exhibits some oscillation which occurs due to instability of the system when large values of Re
are used (Fig. 6.4).
8. Conclusions. The finite element method is computationally satisfactory for solving nonlinear transient problems in two-dimensions (and in one dimension where another dimension was assumed so that the triangulation of the region is possible). The general nonlinear diffusion^convection (Burgers') equation is a good model that represents both parabolic and hyperbolic problems and possesses exact solutions for many combinations of initial/boundary conditions. Comparison of the F.E.M. solution with the exact solution and other numerical solutions shows it to be competitive. The success of the method in solving eqns. (2.1) — (2.3) would make it very attractive to solve
A Finite Element Approach to Nonlinear Transient Problems 227 eqns. (2.4)—(2.5) in future work. Moreover, it is worthwhile to investigate some extensions of the models presented for the non-homogeneous or the two/ three dimensional versions.
REFERENCES
1. A.A. Al-Kharafi and D.J. Evans, The finite element solution of a quasi-linear elliptic equation governing non-Newtonian flows through rectangular ducts , Appl. Math. Modelling 9 (1985), 253-262.
2. W.F. Ames, Numerical Methods for Partial Differential Equations, 2nd ed., Academic Press, New York, 1977.
3. H. Bateman, Monthly Weather Rev. A3 (1915), 163. 4. J.M. Burgers, A mathematical model illustrating the theory of turbulence,
Advances in Applied Mecframes , Vol. 1, p. 171, Academic Press, New York, 1948.
5. J. Caldwell, Numerical approaches to Burgers' equation in numerical methods for nonlinear problems, Vol. 2, Proceedings of the International Conference held at Universidad Poletecnica de Barcelona, 1984 , Pineridge Press, Swansea, U.K., 700-717.
6. J. Caldwell and P. Wanless, A Fourier series approach to Burgers' equation, J. Phys. A: Math. Gen. 14 (1981), 1029-1037.
7. J. Caldwell, P. Wanless and A.E. Cook, A finite element approach to Burgers' equation, Appl. Math. Modelling 5 (1981), 189-193.
8. V. Casulli, R.T. Cheng and U. Bulgrelli, Eulerian-Lagrangian solution of a convection dominated diffusion problem in numerical methods for nonlinear problems. Vol. 2, Proceedings of the International Conference held at Universidad Poletecnica de Barcelona, 1984 , Pineridge Press, Swansea, U.K. 962-971.
9. J.D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Q. Appl. Math. 9 (1951), 225-236.
10. J. Donea, H. Laval, S. Giuliani and L. Quartapello, Taylor-Galerkin method for time-dependent transport problems in structural mechanics in reactor technology, Vol. B; Thermal and Fluid/Structure Dynamics Analysis, (T. Belytschko and J. Donea, Division Coordinators), 1984.
11. D.J. Evans and A.R. Abdullah, The group explicit method for the solution of Burgers' equation, Computing 32 (1984), 239-253.
12. C.A.J. Fletcher, Burgers' equation: A model for all reasons, Proceedings of the 1981 Conference on the Numerical Solution of Partial Differential Equations, Melbourne University, Australia, North-Holland Publishing Co., 1982.
13. E. Hopf, The partial differential equation u + uu = \iu , Comm. Pure Math. 3 (1950), 201-230. t x xx
14. B.L. Lohar and P.C. Jain, Variable mesh cubic spline technique for N-wave solution of Burgers' equation, J. Comput. Phys. 39 (1981), 433-442.
15. H. Nguyen and J. Reynen, A space time finite element approach to Burgers' equation, Numerical Methods for Nonlinear Problems, Vol. 2, Proceedings of the International Conference held at Universidad Poletecnica de Barcelona, Pineridge Press, Swansea, U.K., 718-728.
16. A. Ogata and R.B. Banks, A solution of the differential equation of longitudinal dispersion in porous media, U.S. Geol. Survey Professional Paper, 411-A (1961) .
17. R.D. Richtmyer and K.W. Morton, Stability Studies for Difference Equation: (I) Nonlinear Instability, (II) Coupled Sound and Heat Flow, Report No. NYO 1480-5, Courant Institute of Mathematical Sciences, New York University, New York, 1964,
228 A. A. Al-Kharafi and D. J. Evans 18. K. Sepehrnoori, G.F. Carey and R. Knapp, Convection-diffusion computa
tions, F.E.M. in Engineering, Proceedings of the Int. Conf. on F.E.M., 3rd Univ. North South Wales, Australia,1979, pp.647-656, Unisearch Ltd., Kensington, NSW, Australia.
19. G. Sewell, TWODEPEP, a small general purpose finite element program, IMSL Technical Report Series No. 8102, May 1981.
20. B.D. Sleeman and E. Tuma, On exact solution of a class of reaction-diffusion equations, IMA J. Appl. Math. 33 (1984), 153-168.
21. J. Topper and T, Kawahara, Approximate equation for long nonlinear waves on a viscous fluid, J. of the Physical Society of Japan 44 (1978), 663-666.
22. E. Varoglu and W.D.L. Finn, Space-time F.E. incorporating characteristics for the Burgers1 equation, Internat. J. Numer, Methods Engrg. 16 (1980), 171-184.
23. H.F. Wang and M.P. Anderson, Introduction to Groundwater Modelling, Finite Difference and Finite Element Methods, W.H. Freeman and Company, San Francisco, 1982.
24. O.C. Zienkiewicz, The Finite Element Method, 3rd ed., McGraw-Hill, London, 1977.
NUMERICAL SOLUTION OF MULTI-DIMENSIONAL STEFAN PROBLEM
M. M. Y. ANWAR and A. H. A. AL-HAMMER Mathematics Department, Kuwait University, P.O. Box 5969,
Kuwait
Abstract, A combination of the method of lines, the iterative alternating directions implicit technique and the invariant imbedding approach is used to obtain the numerical solution of a multi-dimensional free boundary problem namely the two dimensional Stefan problem. The analysis of the iterative algorithm is presented and numerical results for a practical situation are given.
1. Introduction. Free or moving surface problems are those problems where the governing equations must be solved subject to certain boundary conditions specified on an a priori unknown surface. Hence the determination of such surface is itself part of the solution of the given problem. Many applications lead to such free boundary value problems, for example: wave propagation, flow through porous medium, the ablation process of a solid and many others.
The present paper is concerned with a specific example namely that of the ablation process of a solid. Such applications can lead to a two dimensional free boundary problem of the Stefan type. There are numerous methods for the numerical solution of such problem. For example, there are methods that use a specific heat capacity to represent the latent heat phase change [1 ,2]; others are based on invariant imbedding approach [3,4,5] , still others use the so-called freezing index [6,7] and solve a variational inequality [8,9]. The approach suggested in this paper is based on the invariant imbedding technique used in [4,5] combined with the alternating direction implicit technique [10,11].
Recently the combination of the well known method of line and invariant
229
230 M. M. Y. Anwar and A. H. A. Al-Hammer
imbedding formalism based on a special alternating direction algorithm namely the fractional steps splitting of the governing equation have been applied in [3,4,5] to obtain numerical solution of the two dimensional free boundary Stefan problem with linear and nonlinear source terms. In [5] an analysis of the multi-dimensional invariant imbedding is given. In this paper we suggest a method that combines the by-lines approximations, the invariant imbedding technique in a way similar to that presented in [3,4,5] but we use a different alternating direction formalism. The analysis of the method follows the same lines of [5] and numerical experiments are performed on different problems and results are compared with those obtained by other authors.
2. Statement of the First Problem. The mathematical model governing the ablation of a solid occupying a time dependent domain D(t) with boundary dD
consisting of two parts 8/λ and 8Z?2 can be expressed as a two dimensional Stefan problem subject to free boundary conditions specified on the free part of the boundary <$D(t) to be denoted by dD2(t) . The fixed part of the boundary is denoted by 3Z)1 . Following [3] this model is written as
| i | + ^ _ | ^ = / ( x 5 Z / j t ) , (x9y)ED(t) (2.1a)
subject to boundary conditions
u= gx,y,t) , x,y)£dDin[x = 0Oy = 0] (2 .1b)
| ^ = 0 , (xiy)€dD1 n [ x = f u 2 / = y] (2 .1c )
u = 0 Λ , , _ _ ... ( 2 . I d ) [x9y) edD2t)
with the initial conditions u(x9y90) = uQ(x9y) €0(0) (2.If)
and D(0) to be given. The domain D(t) c: (0, X) χ (0,?) where X and Y are the upper bounds of
x and y respectively. Figure 1 gives a graphical description of the above problem. It is assumed that the free boundary can be expressed as x= sy, t) and has the inverse representation y=s(x,t) which implies the parametric representation
ZD2t) = sy9t) , y]
or dD2(t) = x9 sx9t) .
Numerical Solution of Multi-dimensional Stefan Problem y
231
D(t)
FIG.l Graphical description of the domain under consideration
2.1 The algorithm. The numerical algorithm suggested in this paper is based on the combination of the method of lines and the invariant imbedding approach outlined in [3] together with the alternating direction implicit formalism proposed in [10, 11], This reduces problem (2.1) into two boundary problems involving the solution of ordinary differential equations subject to the corresponding free boundary conditions. In the first problem we leave the space variable x as a continuous variable and use the method of lines approximation for the space variable y. The time derivative is approximated by the usual backward difference approximation. This problem is to be solved for the time interval
4-0 L - _L Δ^ tt ^tn , τη + -j-
where At is the time step. The resulting problem is as follows
u". + U
u . . — 2u . + u . . u . - u . - ^ ΰ- J^± = f . ( X 9 t x l ) + - ^ L
0 n+\' subject to Ä j / ' u ·-* (Δ*/2)
(2.2a)
(2.2b)
232 M. M. Y. Anwar and A. H. A. Al-Hammer
u'AX) = 0 (2.2c) V
u.(S.) = ° (2.2d) 3 3
s ; (V=-^ i (v^ ' iS · *η*ι) (2-2e)
where u . = u(x9y . , £ , , ) , ü. = u(x,y . ,t )
and the dash is used to denote differentiation with respect to the variable x.
Similarly for the subsequent time interval
te *η + ΊΓ· ^ + Δ* we have similar equations where the space variable y is kept continuous while discretization is made with respect to x and t. Namely we have
+ ^ ± i 1 ! Z i = / ( y f t ) + 7 L _ L (2.3a) ^ kx2 ^ n + 1 (At/2)
sub jec t to
"0(2/) = 90(y>tn+l) , ^ ( 0 ) = 9 ^ ( 0 , * n + 1 ) (2 .3b)
w^(7) - 0 (2 .3c )
w . ( s . ) - 0 (2 .3d)
/ s - £ ~ ~ ^ \ u.(s.)=—q„[x.,s., , t , , , (2 .3e )
where the dot represents differentiation with respect to y and
«* = «(**.*„+!) , β ^ β ^ , ί ^ ) . Thus along each line y-y . the solution jw . , s . of the multi-point free
«7 J 3 boundary problem (2.2) must be found for the time interval from £ to t , ·, . ^ K n n + \ Then subsequently for the time interval from t . , to t , . the solution
^ J n+J n+l u . , s . for t he second m u l t i - p o i n t f r e e boundary problem (2 .3 ) a long each
is 1,
line x = x. must be found. The next step is to apply the invariant imbedding approach used in [3]
to each locally one dimensional problem. Accordingly we consider the Riccati transformation:
Numerical Solution of Multi-dimensional Stefan Problem 233
u. = R(x) u'.+ W. (2.4)
for the first problem. Following the invariant imbedding formalism we substitute ( problems stitute (2.4) into (2.2a) where R and W. are found from the initial value
0
%'
^ - ' - ( ^ έ ) · * 2 · m=° \hy2 tst) J \ät Δζ/2
M o ) = s.(o9t ^ , ) .
(2.5)
f.[x9t , ! ) 1
(2.6)
The two initial conditions in (2.5) and (2.6) are results of (2.4) and (2.2b). The solution of equation (2.5) is given explicitly as
R[x) = L tanh /-Ar + — - x . (2.7)
The numerical solution of (2.6) can be obtained by using a suitable ordinary differential equation solver such as the fourth order Runge-Kutta method. It should be noticed that (2.6) involves u. that is the solution along a previously swept line y = y ._ . It was found very numerically satisfactory to
3 l . . (0) use a Gauss-Seidel type of iteration. Hence if a guess u . is supplied
V then equation (2.6) for the iterative step k can be conveniently written as
ü(k) + ü(k-i)
W( = - (-^+—) -R-W. + R (—u. + ^ °— f.(x,t ^))9 3 \Δζ/2 Lt) 3 W ΰ Ay2 V n + iV
5.(0) = g.0,t ) . (2.8) J d n~ 2.
When applying the Runge-Kutta method data between mesh points is obtained by linear interpolation. Once the values of R and W. at points along the line
V y=y- are found from (2.7) and (2.8) respectively the position of the free
d ^ surface along that line (x = s .) is obtained by substituting (2.4) into (2.2e)
d and using the condition (2.2d). The resulting equation is a nonlinear one and its solution is done iteratively to obtain s . . We have
J ~(k) _-
(at/2) For our purpose it is sufficient for the moment to interpolate between successive points along the line y~y- between which Φ. given above changes sign.
/ \<7 - 3 If no root is found we put s\ J - X. It should be mentioned that over relaxa-
J tion procedure can also be used but we did not attempt to apply it for this problem. Next we solve the two point value problem
3 3\ 3 / 3 l \ 3 3 lM-n\ n+ 2
234 M. M. Y. Anwar and A. H. A. Al-Hammer
ÜW + ^ 1 0 izL=f ,Xtt ) + JL L ( 2 . 1 0 ) 3 M,2 3 n 2 (Δί/2)
subject to (2.2b) and (2.2e) at both ends which are now known. This process is repeated for all the lines y-y.y j = 1 , 2 , . . . , /!/ where y = Y,. It is evident that along y = 0 for j = 0 the solution is given by the first equation in (2.2b).
Secondly the corresponding equations resulting from the application of the invariant imbedding to the second problem (2.3) are treated in exactly the same manner described above for problem (2.2). The analogy between the resulting equations and (2.4) —(2.10) is obvious.
At this stage we have two sequences of solutions J~(k) ~(k)\N , \ (k) (k)\M
\u . , s \ > . and <u . , s . r and we say that a new cycle of the iterative process has been completed. Convergence is considered achieved if the maximum relative error taken over both sequences and in both space directions is less than a pre-specified tolerance; otherwise a new cycle has to be started. Once convergence is achieved a new time step is considered.
2.2 Numevieal results. For the above problem we choose f(x ,y ,t) = 0, D(0) = [0, 1 ] X [0, 1 ] - (x,y) : (x - O 2 + (y - O 2 < ( )2 . The source terms are chosen as
1 - x dx 1 " [ ( i - * ) 2 + (i-j/)2]3* dt ' _ 1 - y , dy_
" 2 ~ [ ( i - * ) 2 + (i-.)2] 3 / 2 dt '
As in [4] the elimination of dx/dt and dy/dt leads to the following equations that describe the motion of the free boundary in x and y directions respectively.
dx dt ϊ'+(-)Ί du I i
dx~ + I ( l - x ) z + (1 -2 / ) '
3/2 [<-*>g-(-*>] and
a * - L 1 + v ^ ; j ^ + ! ( 1 _ x ) 2 + l_y)i\ [Xx)^,-^yiy
As for the initial and boundary conditions we take u = 1 on D(0) , u =0 otherwise, and g= 1 . The propagation of the free boundary is displayed in Fig. 2. Numerical results are in good agreement with those obtained in [4] for the same problem.
Numerical Solution of Multi-dimensional Stefan Problem 235
0.8
0.6
0.4
0.2
— Γ " 0.2 0.4 0.6 0.8
FIG. 2 Evolution of the free boundary for the first problem, kx = äy = 0.02, At = 0.002
3. Statement of the Second Problem. In the present case we consider the numerical solution of a two dimensional Stefan problem [4] with a known analytical solution. Let us consider
d2U , d2U du „I , \
dxz 9*Γ dt
u = gx,y,t) , x9y)EZD1(t)
u = 0
Vu ( dx
, x,y)£D2t)
where f,g,g, and g are chosen such that u = t (t-x-y) is the exact solution of the above problem. The free surface is the line t-x-y = 0,
3.1 Numerical results. The evolution of the free surface of the above problem is investigated by the numerical algorithm described in this paper. Fig. 3 shows plots of the free surface. Results are in agreement with the
236 M. M. Y. Anwar and A. H. A. Al-Hammer
0.8 J
0.6 J
0.4 J
0.2 J
0.0 FIG. 3 Evolution of the free surface for the second problem
Δχ = Δζ/ = 0.02, At = 0.05 analytical expressions and with numerical results presented in [4] as well.
4. Analysis of the Method. In this section we consider the question of convergence of the method described in the previous sections. The analysis is carried on following the same outlines presented in [5], The same results are obtained for the present alternating direction formalism. For the sake of simplicity of the analysis we consider the following model problem
du du du dx2 dy2 dt = f(x,y,t)> x,y) €Dt) (4.1a)
u =gx9y,t) , [x,y) £dDl(t)
, [x,y)e dD2(t) . = — - 0 dn
(4.1b)
(4.1c)
And initially at t = 0, u-uQ[x9y) and D[0) is given. Following the method suggested above we obtain the following discretized
version of the model problem (4.1)
Numerical Solution of Mult i -dimensional Stefan Problem
ü . . , —2u . + ü . 7,11 _L J + 1 J J - l
Δ^2 ϋ". + —^— * *—- = F . ( x , 2/. , t .19ü.9u.)
where
S i m i l a r l y
^(0) = g(0,yj,tn + i)
ü . ( s .) = w'. ( s .) = 0 , <T J 7 J v 31
u .—u .
(4
(4,
(4
237
.2a)
,2b)
.2c)
?! =f(x,y,,tn + 1) + (Δ*/2)
where
u ·, . — 2u . + u . u.+ = F0[x.9y9t ,, 9u . 9u .) (4 .3a )
Δχ
u^O) = ^(x.,0,tn+i) (4.3b)
^ ( β ^ - ά ^ ) - 0 (4.3c)
u . —ü . F0 = f x .9y9t ΜΊ ) + - ^ — 1 . 2 * * n+lJ
( A t / 2 )
We shall be interested in the analysis of the algorithm for the determination of non-negative solutions of problem (4.1). The corresponding conditions to ensure such solution are given in [5] and are readily extended to the present situation. These conditions are (i) F and g are continuously dif ferentiable on RX u : u > 0 for v= 1 ,2
and dR respectively.
(ii) — L and — — > a > — AQ where λ0 is an eigenvalue of the Laplacian du du ° operator.
(iii) Sup \FV|<°° for v= 1 ,2 where supremum is taken over RX u:u > 0 and Rxu:u>0] respectively.
(iv) gx9y, t) > 0 on dR , t > 0 . (v) maxg(0,y90)9 gx9090)9 -^(Ο,ι/,Ο)., - F2x , 0 , 0 ) > 0 , x € ( 0 , X) ,
2/6(0,7). Under these conditions the following corresponding results [5] are readily obtained. PROPOSITION 1. The fixed boundary problem
ψ -2ψ.+ψ 5 ^ + - ^ i — - a ψ. = α (x) 3 == 1,2,...,/!/ J Δ ζ / 2 «/ c/
ψ .(ο) = ψ0(ζ) = o
φ (jr) = 0
238 M. M. Y. Anwar and A. H. A. Al-Hammer
for a . £ C ( 0 , X ] has a unique solution for sufficiently small tsy . If tJ
a.<0 then this solution is non-negative. U ; —
An analogous result is also valid for the corresponding fixed boundary problem.
.. Φ ί + ι - 2 Ψ ί + *^-i ψ. + - -- α ψ = a. Ay) i = 1 ,2, . . . ,M ^ Δχ2 u i *
Ψ^Ο) = ψ0(ϊ)
Ψ0(2/) 0 , ou€C0(0,l] , α^< 0 . The proof is given in [5], The solutions \u . , s . and w . , s . of problem
J J i t
(4.2) and (4.3) respectively are generated by a Gauss-Seidel iteration process namely — T + T — w . = F . (x9 t , , ) (4.4a)
2<fe)(0) =a-(0,t n + 1 ) (4.4b)
= « > ( * < » ) . = « > ' ( , « > ) - . . (4.4c)
where ~(fe) , ~(fc-l) F™(x>t +1) = f(x,y.,t +1)-—3—- J - 1 / + 1
J V w + i ; J V ^ J w + i y ( A t / 2 ) Ay2
And similarly for the second direction we have
2 1_ Δχ2 + At
(fc) _ *(&) U . = ^ - ( , , t M + 1 ) (4.5a)
« ^ ( O ) ^ . « ) , ^ ) (4.5b)
«?)(^fc)) = i?)(e?))=0. (4.5c) The solutions
i w . , s . > and < w . , s . f I J J J l ι 1 J are found as described in the previous sections.
The existence of a positive solution R and i? of the Riccati transformation of the form (2.4) on (09X] and (0,1] respectively follows from their solutions which are of the type given in (2.7). The other two equations for W. and W. are linear and have bounded solutions for bounded F and F res-pectively, which is guaranteed by the assumptions given above.
~(k) ~ ~(k) It should be mentioned that if s. < X then we set u. ΞΞ 0 on
~(k) — ^ . (k) — ^ Γβ . , 1 and similar extension is used if s. < Y. 1 J J ι
Numerical Solution of Multi-dimensional Stefan Problem 239
PROPOSITION 2. Le£ w ( . 0 ) (0) >0 for_ j = 1 , 2 , . . . , N then βΨ^ > 0 and
ϊιΨ^ > 0 on ( 0 , s ( . f e ) ) fo r j = 1 , 2 , . . . , N and k = 1 , 2 , . . . , .
PROOF. From the g iven assumpt ions on g and F, i t fo l lows t h a t
W'HO) =g(0,y.,t M + 1 ) > 0 and # " ( 0 ) > 0 . From the governing equation of W. it follows that W .> 0 on (0,s. ).
. ' U ,-. o r nl o f i ' i Tü r r n ni m um iJ t~ Ύ* H If w. has a relative minimum at x* € (0 , i ) then w'.(x*) = 0 and hence we get u. (χκ) = fv7.(x*) > 0. An analogous result is also valid for
(k) (k) ^ ^ u . and s^ for their corresponding problem. It can be easily observed that if üS. ' — u. > 0 for all j and k preceeding the calculations along
d 3 the line y = y in iteration step 1 then Jm
F(l)(x9t x l ) -F(l 1(x,t A l ) < 0 . 2
This is easily established by recalling the definition of F and noticing Ml 3u I
;(0) = 0. S( 0 )=0 for ,-=1.2 N. Thpn r ^ W ^
that 2 . . . -T— . Similar observation is also valid for the second problem.
PROPOSITION 3. Let u. = 0 , s . = 0 for j = 1 , 2 , . . . , N. Then uv ' > u v
and s > s
m m
PROOF. For t - 1 the result is immediate. Suppose that the assertion is true for all 1 and m preceeding the calculations along the line y = y. and
3 i t e r a t i o n k. Then (wM-W«-1*)' = - M - + ^1 *(*<*> -W-'A-R . (*<*> - ^ " ^ W J / LÄ2/2 AtJ V 3 3 J \ 0 3 J
\ 0 3 I V
Hence by the above observation we get W. ' > W · and hence £?__. > s . ^ ^~(k) ~(fc-i) ^ ^ Furthermore the maximum principle leads to u . > u . . Similarly an
3 3 analogous result for u. and s. can be concluded. The previous results show that the sequences
f ' . f - "?>.·?' have monotonely increasing lower bounds. Similarly it can be shown that they also have monotonely decreasing upper bounds.
Let Γ . = ü . + max q (x , y , t . 1 ) ,
3 3 a/? n + 2 let L be the operator acting on w. in (4.4a), then
3 PROPOSITION 4. Le£
y(.0) = f. , s(.0) = x . 3 3 3
r ~ ( k ) ~'(/c)i Then the Gauss-Seidel iterates 1 i/. , S. satisfy
_ l 3 3 J — ^
240 M. M. Y. A n w a r a n d A. H. A. Al-Hammer
£/<*>< u * * - 1 > and S(k)<s[k'^ 3 3 - — 3 3 /
The proof is done by induction and using the maximum principle for the operator L . PROPOSITION 5. For the sequences u. and u\ we have
3 3 0 < uk) < U(.k) < f . .
3 3 3
PROOF, Since s . < £> . and u . < &. then by inductive argument and J J 3 3
using result 3 we can directly prove the above statement. Similar result holds for the second problem.
(fc)i Finally this last result shows that for each j the sequence u . 3
is a uniformly bounded sequence. Now we can state the following result. PROPOSITION 6. The sequence u. , s. converges to a solution 1 3 3]
PROOF. The equation
J J where L is the operator in (4.4a) implies that u. is uniformly bounded on
(k) (k) - ^ (/c)i (k)' (0 , s: ) and by extension on (s. , X). Thus both u. \ and u . ) are
3 φ 3 # J ~(&) ^ sequences of uniformly bounded equi-continuous functions so ü . converges
. «-7 monotonically to a continuously differentiable function ü. as k-*°°, and at the same time the monotone sequence s . converges to a limit s* .
Similar argument can be used to show that \U . ,S. converges to U*,S*. 1 3 JJ
To guarantee the uniqueness of solution [5] we impose the restriction
3 + 1 3-l > 0 o n [ r -χ]_
Similarly we establish similar results for the second problem under the corresponding restriction
u · . + u . Ί F _1±1 lz± > o on [S*,J]. 2 . 2 ^
5. Conclusion. In this paper an algorithm is suggested for the numerical solution of multi-dimensional free boundary Stefan problems. The method is based on the combination of the method of lines and invariant imbedding [5] using the alternating direction implicit iterative formalism. The formulation of the resulting problem falls within the frame of the class of problems considered in [5] which leads to adopting similar arguments for the analysis of the method. Numerical experiments using the suggested method are in good agreement with previously obtained results.
Numerical Solution of Multi-dimensional Stefan Problem 241
REFERENCES
1. C. Bonacina, G. Comini, A. Fasano and M. Primicerio, Numerical solution of phase change problem, Int. J. Heat Mass Transfer 16 (1973), 1825-1832.
2. G. Comini, S. del Guidice, R.L. Lewis and P.C. Zienkiewicz, Finite element solution of nonlinear heat conduction problems with special reference to phase change, Int. J. Numer. Methods Eng. 8 (1974), 613-626.
3. G.H. Meyer, An alternating direction method for multi-dimensional para -bolic free surface problems, Int. J. Numer. Methods Eng. 11 (1977), 741-752.
4. G.H. Meyer, Direct and iterative one-dimensional front tracking methods for the two-dimensional Stefan problem, Numer. Heat Transfer 1 (1978), 351-364.
5. G.H. Meyer, Free boundary problems with nonlinear source terms, Numer. Math. A3 (1984), 463-482.
6. J. Aguiree-Puente and M. Fremond, Frost propagation in wet porous media, in Lecture Notes in Mathematics, vol.503, Springer-Verlag, Berlin, 1976, pp.137-147.
7. M. Fremond, Methodes Variationneil es en calcul des structures, Cours de D.E.A., Ecole Nationale des Ponts et Chaussees, Paris, 1980.
8. Y. Ichikawa and N. Kikuchi, A one-phase multi-dimensional Stefan problem by the method of variational inequalities, Int. J, Numer. Methods Eng. 14 (1979), 1197-1220.
9. N. Kikuchi and Y. Ichikawa, Numerical methods for a two-phase Stefan problem by variational inequalities, Int. J. Numer. Methods Eng. 14 (1979), 1221-1239.
10. D.W. Peaceman, and H.H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Applied Math. 3 (1955), 28-41.
11. J. Douglas and J.E. Gunn, A general formulation of alternating direction implicit methods, Part I, Parabolic and hyperbolic problems, Numer. Math. 6 (1964), 428-453.
MAA—Q
A SELF-CONSISTENT FINITE-DIFFERENCE TREATMENT OF THE MOVING BOUNDARY
FOR LONG WAVES IN SHALLOW WATER
A. CHALABI Institut de Mathematiques, Universite de Constantine,
Constantine, Algeria
Abstract, In this paper we deal with the numerical solution of a moving boundary value problem related to long waves in shallow water. The algorithm is based on the finite difference method. The major novelty in the present study lies in the treatment of the boundary. Some numerical results are given at the end of this paper.
1. Introduction. We give in this paper a numerical solution of the system of Saint Venant, in the unidimensional case, for a parabolic domain containing a liquid provided with an oscillatory movement. The choice of the domain is justified by the existence of an exact solution. Such a solution allows us to deduce the boundary value which we impose at the boundary for finding again numerically the solution of the problem. Besides, the exact solution will be useful for comparison of the calculated values of the solution.
In [3] the problem was studied, but in that paper the authors used a homogeneous boundary condition. In our present paper we use a nonhomogeneous boundary condition taking into account the movement of the boundary. This condition is more realistic if one considers the displacement of the boundary.
The numerical treatment of the system of long waves near the boundary is not easy, since the system is not valid at the boundary, however the use of a "good" boundary condition leads to better results.
We give some illustrations using numerical examples. The same problem with a fixed boundary in the bidimensional case was studied in [2] by using the finite element method,
243
244
N o t a t i o n s :
A. Chalabi
Amplitude of the horizontal oscillations Abscissa of the point of maximal depth Acceleration due to gravity Depth of the liquid at the centre of the free surface at the equilibrium position Depth of the liquid Half of the width of the free surface at the equilibrium position Velocity Component of the displacement Constant related to the form of the domain [g a) 2 : frequency of the oscillations Abscissa of the boundary Velocity of the boundary Initial velocity
2. Study of the Unidimensional Case. The system of Saint Venant in the unidimensional case in a parabolic domain is
dh du du — + u — + dt dx -gax = 0 ,
dh , dh . 7 du — +u — +fc — = 0. dt dx dx The general solution calculated in [1] is:
X = A sinu)t
- I/Ji fx-Asinut\2\ ~Μ\ V I Jl9
(2 .1 )
(2 .2 )
( 2 . 3 )
( 2 . 4 )
but u — — , thus dt
u = 4ω cos wt . ( 2 . 5 )
2.1 The Boundary Conditions, The boundary of the domain cor responds to the p o i n t s where h = 09 then from (2 .4)
x-X = ± L . ( 2 . 6 )
Let us denote by x-,, Χτ, the abscissa and the displacement of the boundary,
Fini te -d i f fe rence T r e a t m e n t of t h e M o v i n g B o u n d a r y 245
then xh-x
h= ±L . ( 2 . 6 ) '
If we differentiate (2.6)' with respect to the time, we get ax-, dX-
dt dt thus
dx
dxn dX-, h b = 0 , (2.7)
b dt
(2.8) dX where u-, is the velocity of the boundary since u— -=— from the hypothesis o dt
made in [1]. The abscissa a\ will be calculated every time we have 7z=0. In the numerical solution of the system (2.1) —(2.2) we calculate h at
the instant t, using the values of u at the instant (t — Δ £ ) , then we determine the abscissa x, of the boundary at the instant t . In the calculation of u at the instant t , we use h and U-, .
3. The Algorithm of Solution. We solve the problem in the half straight line x > 0 , and for that we consider a horizontal translation of the parabolic domain, with an amplitude b* The system (2.1) —(2.2) becomes:
du . du . dh , i , \ r, /o i \ irr* u—+ g — + gax-b) = 0 (3.1) dt dX dX oh , dh , 7 du Λ /0 ON — + w — +n— = 0 . (3.2) dt dX dX
Then the general solution given in [l] will be
X = A sin out (3.3)
h = ^]~(X~bL~XJ - (3.4)
The boundary condition remains unchanged, the constant A may be determined from the initial velocity uQ = ω · A . Outside the domain we put u = h = 0 .
3.1 Treatment of the Boundary. (a) The abscissa X-, of the boundary is determined by interpolation using either the values of h. and h. , satisfying 7z . < ε and /z . . > ε (for
τ ^ +1 ^ ^ + l the first boundary) or the values h._, and h. for which h._1> ε and /z . < ε (for the second boundary), (ε = 10 in the numerical examples). (b) The velocity u-, of the boundary is calculated at every step of the time by discretization of the relation (2.8) that is: u-, = (Δχτ)/Δΐ but since the point for which the abscissa x, is not necessarily a node point of integration, we determine u> at the node closest to the boundary inside the
246 A. Chalabi domain h. >0. The movement is assumed to be quasi-rigid and the velocity depends only on the time, then we can put u.= u,.
(c) For the calculation of h at a point at the instant t, we have to distinguish if the point is dry or wet at the instant (t —At). If the point is wet, there is no problem, but if the point is dry, that is if the liquid goes up, we need the value of h at this point at the instant (t—At). We calculated this value by linear extrapolation using the two points which precede it and are inside the liquid domain. The formula giving h at the instant t is:
h.(t) = [^(t-At)]extrap.
-iH^-^H^-^Urap.-V^-^H-lt*-^ (3·5) at every step of the time, we put h= 0 at each point outside the liquid do~ main; we do this by testing if the point is inside or outside the domain.
3.2 The Numerical Scheme, For the numerical solution of the system (3.1) — (3.2), we used a variant of an algorithm proposed by Flather and Heaps [3], These authors put as boundary condition u= 0 but at the same time they assume that the boundary is moving. This situation does not correspond to physical reality, that is why we introduced a nonhomogeneous condition u—u-,
which is more realistic than the homogeneous condition.
3.3 Details of the Scheme,
(a) The convective terms are treated by the angular derivative method [7], where the spatial derivative is approximated by centred difference in space and time. The sense of the integration is inverted at each step of the time. (b) The variable coefficients of the derivatives are calculated using the values we got at the preceding step. (c) The stability of the scheme is controlled by the CFL criterion! [4],[6]);
1ST < A x . (3.6) STgJi
max (d) Calculation of h,
h .(t+At) = V t ~Tx hi^ui^-hi-iW W t ) · ' 3 · 7 )
The formula (3.1) is used for the points which were wet at the instant it —At) and were in the liquid domain. For the other points of the liquid domain we use Section 3.1 (c).
After the evaluation of h, we determine the new liquid domain by calculating the new abscissa of the boundaries. (e) Calculation of u. For the points which are not the extreme nodes of
Finite-difference Treatment of the Moving Boundary
the domain, we use the following formulas:
( i ) For the odd step of time and for increasing i :
Ι ^ ( £ + Δ £ ) - ^ ( £ ) | / Δ £
= i ^ U ) f ^ l ( t ) - ^ ( t ) + ^ ( t + A t ) - ^ - l ( t + A t )
with
M T ( ^ ) = ^ ( u . ( t ) + 2u.t) +u. At)).
(ii) For the even step of time and for decreasing Ί :
[u^t+bt)-TTAt)
247
u.t))/Lt
~At) r
-ghi+l t+^-h^t + ät)/bx-goLix^-b) ,
(3.8)
(3.9)
(3.10)
the extreme nodes are treated by the method described in Section 3.1(b). We consider the inversion of the sense of integration for a stability reason.
4. Numerical Results
FIG. 1 Number of nodes of integration = 48 H = 0.1m A = 1 m t = 50 Δ£
exact value calculated value
248 A. C h a l a b i
F I G . 2
H = 2m A = 1 m
At = 0.5 seconds t = 50 At
u 1
i -
k
_ . J .. , ^ > > I · > 1 1 t I 1 1 1
i . I . . 1 1 . i . . . - a
FIG. 3 # = 2m A = 1 m At = 0.5 seconds t = 50 At
5. Conclusion. The description of the moving boundary requires a rigorous treatment since a bad determination of the boundary affects the conservation of the liquid and leads to wrong results.
The numerical examples show us the existence of some fluctuations at the boundary; this is a natural difficulty because the model used is not valid at the boundary, nevertheless inside the domain the results are accurate enough.
REFERENCES 1. F.K. Ball, An exact theory of simple finite shallow water oscillations
on a rotating earth, Proc, 1st Austral, Conf. Hydraulics and Fluid Mech,, (1962), Pergamon, 1964.
2. A. Chalabi, Analyse numerique du probleme des ondes longues en eau peu profonde, Calcolo (3) 19 (1982), 269-288.
3. R.A. Flather and N.S. Heaps, Tidal computations for Morecambe bay, J. Roy, Astr. Soc. 42 (1975), 489-517.
4. N. Gastinel, Introduction elementaire aux methodes numeriques pour la resolution de problemes d'equations aux derivees partielles, Cours INPG-USMG, Grenoble, 1978.
Finite-difference Treatment of the Moving Boundary 249
5. S.H. Lamb, Hydrodynamics, Cambridge University Press, 1932. 6. T.D. Richtmyer and K.W. Morton, Difference Methods for Initial Value
Problems, John Wiley and Sons, New-York, 1967. 7. K.V. Roberts and N. Weiss, Convective difference schemes, Math. Comp.
20 (1966), 272-299. 8. A. Sielecki and M.G. Wurtele, The numerical integration of the nonlinear
shallow water equations with sloping boundaries, J. Comp. Physics 6 (1970), 219-236.
9. J.J. Stoker, Water Waves, Interscience Publishers, New York, 1953.
RATIONAL AND POLYNOMIAL APPROXIMATIONS FROM
ULTRASPHERICAL SERIES FOR LINEAR DIFFERENTIAL EQUATIONS
E. H. DOHA Department of Mathematics, Faculty of Science, Cairo University,
Giza, Egypt
Abstract, In this paper a method is given for obtaining simultaneously the rational and polynomial approximations for function defined by linear differential equation with its associated boundary or initial conditions. The method depends basically on that an expansion in a series of ultraspherical
(a) . . . . . . . polynomials C for the function and its derivatives occurring in the differential equation is assumed; the coefficients of expansion are then determined by substituting in the differential equation and equating the coefficients. This expansion for the function enables us to get the sought--for rational and polynomial approximations for any possible non-zero value of the parameter a.
The method in its present form may be considered as an extension of Doha's method [1984] and Elliott's method [1960] into the complex domain. 1. Introduction. The well-known effective means of producing rational approximations to a function y(z) in a complex variable is to develop elements of Pads' table from its Taylor series. This table is a two-dimensional array whose (m,n) element is defined as that rational function of degree m
in the numerator and n in the denominator whose own Taylor series oo
y c zr
r=0 +
say, agrees with that of y (z) up to and including the term in z . Discussion of the Pads' table from a numerical point of view is discussed in Handscomb [l],
In fact, rational approximations whose numerator and denominator have the same or nearly the same degree generally produce approximation results
251
252 E. H. Doha superior to polynomial approximations for the same amount of computation effort, see for instance, Ralston and Rabinowitz [2].
In Doha [3], a method for obtaining simultaneously polynomial and rational approximations from Chebyshev and Legendre series for function defined by linear differential equation with its associated boundary conditions has been described. The essence of the method is that an expansion in either Chebyshev or Legendre polynomials is assumed for the function and its derivatives occurring in the differential equation; the coefficients of expansion are then determined by substituting in the differential equation and equating the coefficients.
Our principal aim in the present paper is to give an extension of Doha's method; but the function y(x) and its derivatives are expanded in ultra-spherical polynomials.
The ultraspherical (Gegenbauer) polynomials associated with the real (a) parameter a > -\ are a sequence of polynomials C (x) (7-2 = 0, 1,2,...),
each respectively of degree n, satisfying the orthogonality relation 1 I (\~x2)a~l C^\x)cf\x) dx = 0 . (τηΦη) (0 -1
Any function y (x) with 1 (i -x ) a 5 yx) dx < «>
-i has an orthogonal expansion in such polynomials
CO
»ω= Σ V»a)<*> (2)
n=o whose coefficients are given by
1 X 2 %=\ (i-x2rly(x)c(
na)(x)dx /\ ( 1 _ ,2 f - i | c (a ) ( x ) J dXm
-1 -i For our present purposes it is convenient to standardize the ultra-
spherical polynomials so that C(a)\) = Γ(η+2α)/Γ(2α) η\ (4)
which is the usual standardization (see, Abramowitz and Stegun [4]). In this form the polynomials may be generated using the recurrence formula
(3)
( n + l l f i W = 2(n + a)xC^a)(x)-(n + 2a-\)C^1(x) (n = 1 , 2 , . . . )
(cO (oO s t a r t i n g from C^. [x) =1 and C7 (x) = 2ax, or ob ta ined from Rodigue ' s formula
(5)
Rational and Polynomial Approximations 253
Nn „, , „ v ~, , i v , 7n C(a)x) = ( - 0 Γ(η + 2α) Γ(α+|) ( ] _ χ 2 ) ^ - α ^ n 2nn! Γ(2α)(η + α + | ) d*n
l - x 2 f + ^'i (6)
Of these polynomials, the most commonly used are the Chebyshev polynomials T (x) of the first kind, corresponding to a = 0; the Legendre polynomials P (x) for which a=\ ; and the Chebyshev polynomials U (x) of the second kind (a= 1). In the first case the standardization is different from that given in equation (6), since
T (x) = % lim I C(a)(x) . (7) a-* 0
The Legendre polynomials P (x) and the Chebyshev polynomials U (x) are obtained directly from equation (6) by substituting a = J and 1 respectively. From the orthogonality property of the ultraspherical polynomials, the coefficients a of (3) takes the form
n a =η!(η + α)Γ(α)Γ(2α1 Γ (, _χ2) α-J (α) χ) υχ)άχ # ( g )
η /π Γ(η + 2α) Γ(α +J) _^ η
In general, it is not possible to evaluate the integral occurring in (8) explicitly, and to find a , recourse has to be made to suitable quadrature technique.
The present method enables us to find these coefficients directly, provided that the function y Or) should satisfy a linear differential equation with appropriate boundary conditions. The solution of linear differential equations in series of Chebyshev polynomials T (x) has been given by Lanczos [5], Clenshaw [6] and Fox [7]. An extension of Clenshaw's method has been described by Elliott [8], and an extension of Clenshaw's method into the complex domain has been given in Doha [3].
The proposed method may also be considered as an extension of Elliott's method into the complex domain, which consequently yields rational approximation and as a special case the polynomial one. No approximate formula are used, so that errors in the solution are caused only by the accumulation of rounding off errors. This can always be kept within assigned limits by retention of a number of guarding figures. Also, the method may conveniently be used with either a desk calculator or an automatic computer.
The method is described in sections 2 and 3, and is illustrated by numerical example in section 4. Some concluding remarks are given in section 5.
254 E. H. Doha
2. Solution of Linear Differential Equations in Ultraspherical Polynomials. Suppose we have a linear differential equation of order m of the form
m yz-) Pi(x) ^£=q(x) , (9) i=0 dx
where q(x) and P. (x) , t = 0 , 1 , 2 , . . . ,m are functions in x. The complete system determining uniquely the function y(x) needs m initial or boundary conditions together with (9). Then if y (x) is continuous in the closed interval [-1,1] , we can write
oo
n= 0 where the coefficients a are to be determined. The kth. derivative of y(x)
n ΰ
is expanded formally by
.<« (*)= y a^k)C(n
a)(x) * = 0,l,2,...,ra . (11) n= 0
The method of determining the coefficients a depends upon the following n (et)
two recurrence relations for the ultraspherical polynomials C (x) , namely d C (<*. (x) d C(CL\ (x)
and 2(n + a)xC(
na)x) = (n + 1) C^x) + (n + 2a- 1) ( x ) , (13)
both of which are valid for n > \ . From (11) we may write
yk+l\*)= I *(n
k+l)c^(x) n= o
and making use of (12) gives
/ * + 1 ) (x) = £ (fc+i) /7(^+1) ?2-l « + 1
L 2(rc + a-i) 2(n+a+i) d C(a)(x)
n dx n= l
On differentiating equation (11), we find oo 3 r(ot) , x
(fc+l), Λ V (fc) α w W
* (" = L % & n= o
from which, on equating coefficients we have (fc+i) (fe+i)
a w = _»z i nn—, n>] , (,4) n 2(n+a-l) 2(n+a+l)
For computing purposes, we define a related set of coefficients b by writing
Rational and Polynomial Approximations
(fc) / A x Ak) aK = (n + OL) bK
n v ' n n > 0 , fc = 0, 1 , 2, . . . ,/n
and accordingly, equation (14) takes the simple form
x ' n n-\ n-i (a) Again, let C (y) denote the coefficient of C (x) in the expansion of y .
Then
255
(15)
(16)
xy = S a xCW(x L n n
n=0 by using equation (13) and rearranging terms, we have
xy -i Thus, n=oL
w-i (η+2α)α n+i 2(n+a-l) 2(rc+a+i)
na . (n + 2a)a _,_, , / Λ n-l , n+l C (#2/) = — r + -^V y / 2(n+a-i) and in terms of the coefficients b
2(n+a+!) we have
c(a)(x)
n > 0 (17)
n / λ η l· i +201 l· n\ #/ 2 n _ 1 2 n+l ' « > 0 (18)
By continued application of equation (17), we can find C (x2y) , C (x3y) etc. So in equation (9), C (p .(x) (d y/dx )) can be written down if p.(x) is a polynomial in x. In cases where p.(x); i =0,l,2,...,m are not polynomials in x it is sometimes best to replace them by suitable polynomial approximations. It is to be noted here that, in equation (18) we may re-
(k) (k) place y(x) by y (x) provided b is changed by b
In general, equations (14) and (16) are only valid for n ^ l , and the use of equations (15) and (17) gives rise to recurrence relations where we might have to assign a meaning to a or i for negative values of n. It is well-known that for the Chebyshev polynomials T (x) , a = a_ , for all values of n ,
Elliott's [8] has reported that if 2 a = m , where m is an integer, (k) then for the coefficients a , we have
(k) a ,' v = —a
with -(m+r)
Ak) _ Ak) a-i - a-z
and for the coefficients b <*>
with h(k) _ Ak)
Ak) -1
Ak)
-(m-l)
-(m-l)
r > 0
v > 0
(19)
(20)
256 E. H. D o h a
and when 2a i s not an i n t e g e r , then
aW = *<*> = 0 -n -n
for n > 1 (21)
The boundary conditions are generally given at either x = 0 or x = ± 1 , (a), n (a), n
The values of C^ (x) at these points are given here
C w ( l ) = Γ(η+2α)/Γ(2α) η\
C^\-l) =(-\f Ca)(\)
c(a)(o) = n odd
L(-Dw/ 2r(ffa) Γ(α)(*)ΐ
(22)
These results are valid for all values of a, except a = 0. Substituting (10) and (11) into (9), we obtain, by means of (16) and
(k) (18) and its continued application, relations for the coefficients b for /c= 0,1,2,. ..,772, for all n . These relations and those obtained from the boundary or initial conditions are equivalent to an infinite system of linear
(k) equations in the unknowns b . The numerical solution of these equations n can be performed by the two methods described in detail by Clenshaw [6], These are the method of recurrence and the iterative method. 3. Extension of the Method in the Complex Domain. To extend the method into the complex domain we consider instead the function y(zx) , where x is the independent variable, -1 < x < 1 , and z is regarded as a parameter which may take any real or complex values. The function y(zx) satisfies the differential equation
77? V L
i= 0
d^yjzx)
z ax q(zx) (23)
The method of section 2 is applied to get a polynomial approximation in x,
say R (z,x), for y(zx) in the form N
Bjz9x) = — ] — Y a z) C(° # a(z) L n n (24) n = o
The coefficients a (z) are now rational functions in z since the recur-n Akh rence relations for the coefficients a (z) involve such functions. The
n multiplying factor l/a(s) is another rational function in z resulting from the satisfaction of an initial condition. Finally, putting x = 1 in (24) yields a rational function R (z,\)=IAz), say, which approximates y(z)
for any real or complex values of z, \z\ < 1 . It is worth to be mentioned that, if we put z=\ in (24) then we get
Rat iona l a n d Po lynomia l A p p r o x i m a t i o n s
usua l polynomial approximat ion for the func t i on y(x) .
257
4. Numerical Example. Suppose we want to find a rational and polynomial 2 approximation for the function e , x € [-1 , 1 ] , by using the ultraspherical
polynomials. This function satisfies the differential equation
dx - 2xy = 0 , i/(0) =1 (25)
If x—^xz , R(z 9x) = y(zx), and - 1 < x < 1 , then equation (25) takes the form
(26) 2z2xR = 0 , R(z>0) = 1 . dx
Also, let R(z,x) be given by equation (10). Comparing the coefficients of (a) C (x) in the expansion of the terms of equation (26), we have
C (RJ) - 2z2C (xR) = 0 .
Therefore, from equation (17), we get
a — z n
n-l (n+2a)a n+ 1
(n+a-l) (n+a+l) = 0
and in terms of the coefficients b , we find n
(n + a)b(l) = z2\nb +(n + 2a)b 1 n [ n-l K ' n+lJ
With t h i s equa t ion i n the form
n-l n b —(n + 2a) b ί Ί (27)
.(1) and equation (16) with k = 0, we can readily compute b and 2? , and hence ( 1 ) 2 n n
a and a . Since ex is an even function, n n
for all n.
The two cases a = 0 and a = \ , which are corresponding to the expansions in Chebyshev polynomials of the first kind and Legendre polynomials, have been discussed in detail in Doha [3]. In the following we discuss the case a = 1 . which is corresponding to the expansion in Chebyshev polynomials of the second kind.
In this case, and if we put a= 1 in equations (16) and (27), we get the two recurrence relations
n-l (1) b K J + 2(n+ 1) b ; and n+l v ' n '
n-l n 2 n v ' n+lj
(28)
(29)
Tabl
e 1.
C
ompu
tatio
n of
Tr
ial
Solu
tion
for
Exa
mpl
e 1
in
Term
s of
a
and
a
usin
g C
heby
shev
P
olyn
omia
ls
of
the
Seco
nd
Kin
d
2 0 1 2 3 4 5 6 7 8 9 10
11
12
Trial
zl2~n a
(z)
n
~*l2
%
*10a
2
z8 a,
26
a c b
zh a Q
z2
aio
a l2
zu
~n
a(
l\z
) n
*">α
™
3
«6 ί>
^ (1)
z α η
2 (1)
z α 9
«ϋ'
1
+383
3856
0 + 38
3385
60
+958
4640
+9
5846
40
+119
8080
+1
1980
80
+ 99
840
+998
40
+ 62
40
+ 62
40
+ 312
+ 31
2 + 13
z2
-958
4640
-119
8080
-998
40
-624
0
-312
-13
a*
+199
6800
+7
9872
0 +1
9968
0 +9
9840
+1
2480
+ 62
40
+ 52
0 + 20
8 + 13
z*
-199
680
-124
80
-520
-13
zQ
+149
76
+ 249
6 +6
24
+ 52
+ 13
z10
-624
-13
z12
+ 13
n
+1.3
2819
2
+0.3
7259
1
+0.0
4824
3
+0.0
0408
9
+0.0
0025
8
+0.0
0001
3
+0.0
0000
1
NOTE:
z^ a
a =
6240 -31
2 z2
+ \
3zh
, a^a^
= 99840 + 208
z*
8 /
Rational and Polynomial Approximations 259 As a starting point we have taken
With these starting values, equations (28) and (29) can be used to compute b , b for all n < 12 . The coefficients a and a can now be computed
n n n n r
by making use of equation (15) with k = 0 and k = 1 respectively. These are given in Table 1. These values of a have to be multiplied by a factor 1 /OL(Z) which is determined from the as-yet-unsatisfied boundary condition. This gives
a(z) = [38338560- 19169280 z2 + 4392960 zh -599040 zS + 52416 zQ
-2912s10 +91 zl2]/z12
Thus we have 6 Rz'x) = ^h Σ y ^ C w · ( 3 o >
n=0 Put x = 1 , then equation (30) yields the required rational approximation for 2
ez in the form 2 e2 = /?(* , 1) = P(z)/Piz) , t = i^X
where P(z) =38338560+ 19169280 s2 + 4392960 24 + 599040 26 + 52416 s8
As, a check we find that i?(l, 1) =2.718282 = e. Meanwhile, the polynomial approximation from the Chebyshev polynomials of the second kind can be obtained from equation (30) by taking z = 1 . In this case
βχ2 = R(\ ,x) = \.328\92UQx) + 0.372591 Uz[x) + 0.048243 Uh(x) +0.004089 U&x)
+ 0.000258 UQx) +0.000013 U1Qx) +0.000001 Ul2[x) .
5 Conclusion. In this paper we have described a method which enables us to find simultaneously the rational and, as a special case, polynomial approximations for arbitrary function f(x) expanded in an infinite series of
(a) ultraspherical polynomials C (x). The coefficients of expansion may be obtained to any required degree of accuracy. The function fix) is assumed to satisfy some linear differential equation with associated initial or boundary conditions. The differential equation can then be solved directly to give the unknown coefficients of expansion.
The rational approximation to be obtained by this method can be considered as an extension of Elliott's method [8] and Doha's method [3] into the complex domain. It is of fundamental importance to note that the polynomial approximation for fx) is obtained directly from its rational one, R(z ,x) , simply by putting the parameter z equal to unity.
No need to discuss the error analysis of this method, because the
260 E. H. Doha
coefficients of the expansion are in general readily evaluated, and approximations of any specified accuracy are provided by mere truncation.
REFERENCES 1. D.C. Handscomb, (Ed.) Methods of Numerical Approximation, Pergamon
Press, Oxford, 1966. 2. A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis,
McGraw-Hill, New York, 1978. 3. E.H. Doha, Rational and polynomial approximation from Chebyshev and
Legendre series for linear differential equations, Arabian J. Sei. Engineering 10 (1985), 3-13.
4. M. Abramowitz and I.A. Stegun, (Eds.), Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series 55, New York, 1970.
5. C. Lanczos, Trigonometric interpolation of emperical and analytical functions, J. Math. Phys. 17 (1938), 123-199.
6. C.W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Phil. Soc. 53 (1957), 134-149.
7. L. Fox, Chebyshev methods for ordinary differential equations, Computer Journal 4 (1961), 318-331.
8. D. Elliott, The expansion of functions in ultraspherical polynomials, J. Austral. Math. Soc. 1 (1960), 428-438.
THE ASYMPTOTIC BEHAVIOUR OF SOME SEMI-DISCRETE AND FULLY-DISCRETE FINITE ELEMENT SCHEMES FOR THE
EVOLUTION NAVIER-STOKES EQUATIONS
W. N. JUREIDINI Depar tment of Mathematics , American University of Beirut, Beirut,
Lebanon
Abstract. The asymptotic behaviour of a semi-discrete scheme and a fully--discrete scheme for the evolution Navier-Stokes equations are considered. These schemes arise from some standard and nonstandard finite element discretizations in the space variable. The fully-discrete scheme is based on the backward Euler method for time discretization. It is shown that the solutions of these schemes remain bounded. Moreover, it is shown that under the well-known sufficient condition for uniqueness of the corresponding discrete steady-state problem, these solutions converge asymptotically to the solution of the steady-state problem. The latter results are the analogues of the results obtained by Payne [3] for the continuous problem.
1. Introduction. Semi-discretizations of the evolution Navier-Stokes equations, in the velocity-pressure formulation or in the stream function formulation for the two dimensional problem, based on standard and noncon-forming finite element methods ([2], [4], [1]), lead to the problem of finding a u: [Ο,οο) ->Μ that satisfies
[u ,v] + v((u9v)) + b(u9u,v) = /»^ ) , for all y in M, (1)
u(0) = v° . (2) M is a finite dimensional subspace (usually of piecewise polynomials) of a normed linear space H with normII «II . [ · , · ] is an inner product on M that gives rise to a norm |·| such that
\v\2 < cx \ ν γ , for all i; in M. (3)
((·>·)) is a bilinear continuous form on M which is coercive, with
261
262 W. N. Ju re id in i
I. I|2
and <(V,V)), for all D in M, (4)
| ((v , w) ) | < c3 || v || || w || , for all u,y in M. (5)
b( · * · 9 ·) is a trilinear continuous form on M with
| b(u ,V , w) | < Q . ||w I ||t>|| | |w|| , for a l l u,V,w i n M, (6)
and b(u,v,v) = 0 , for a l l u , t> in M . (7)
(· 5 « ) : ^ X F ^ i ? , #* being the dual of #, denotes the usual bilinear map with
\g>v)\ < \\ρ\\^ \\v\\ , for all g in H*, v in M , (8)
where || · || denotes the norm on H*. V > 0 is a constant representing the kinematic viscosity, /: [0,«>) -> #*
is given, and w 6M is also given. A fully discrete problem that we shall consider is the fully implicit
t- T«. 0 1 7Π+\ i r\ ^ + 1 · i r-
scheme: When u ,u , . . . , u a r e known, m > 0 , u i s t he s o l u t i o n of
[dum,v] + v((um+\v)) + b(um,um+l ,v) = f ,v), for a l l v in M, (9)
where for a t ime mesh k> 0 , 3w = \/k (u — um) , and (m+l)fc
/"=£ J /(*)<** (10) mk
which we assume exists in H , for all m > 0 . The discrete steady-state problem is expressed by
v(U 9v))+ b U9U9v) =(F9v) 9 for all i; in M, (11) where FEH* is given.
It is known that (see [4]) for FE H*, (11) has a so lution U in M
that satisfies Η < ( ^ 2 Γ Ί Μ Ι * · (·2)
Moreover, U is unique if λ > 0 , where
λ= (ve2f- ejF^. (13) Deriving existence, uniqueness and error estimate results for the prob
lems (1) —(2) and (9) is not our interest in this paper. These results are found, for all the problems mentioned earlier, in [2], [4] and [1], We shall be concerned with the asymptotic behaviour of these solutions.
We show that if / is bounded in H then u(t) and um remain bounded for all t>0 and m>0. Moreover, if f(t) converges to F as t-»<», and
The Evolution Navier-Stokes Equations 263 if the condition λ > 0 of uniqueness holds, then both u(t) and um converge asymptotically to U.
More precisely, our results are stated in Theorems 1 and 2 below. Their proofs are developed in Sections 2 and 3 respectively.
Before stating the results we introduce the space
£"(0,-.;/?*) = ffs[0,»)-»-//*; Hlfflll < - . where
HI? HI =ess sup \\g(t)\\^. t > 0
Our results are:
THEOREM 1. Let / be in L°°( 0 ,<» ; #*) . Then, there exists a constant C>0
such that for any T > 0,
| U ( T ) | 2 « l uY + ^ v ^ r M l l f l l l 2 . (14)
Moreover, if λ of (13) is positive and if for some s , 0 < s < λ/(v c. c ) T
lim f e8(t-T)\\f(t)-F\\ldt = 0, (15)
then lim \u(t) -U\ = 0 . (16)
THEOREM 2 . Let^ / be i n L°°(O s co;#*) . Then t h e r e e x i s t s a c o n s t a n t C > 0
such t h a t for a l l m = 1 , 2 , . . . ,
| M » | 2 < \u\2 + C(ve2yl \\\f\\\2 . (17)
Moreover, i f λ of (13) i s p o s i t i v e , and i f (15) ho lds for some s > 0 , wi th
(esk-\)/k < \/(vc1ez+c^\\F\\l) , (18) then
l im \um-U\ - 0 . (19)
We note that (15) certainly holds for any s if f(t)=F for all t . This will be the case if (9) is to be used as an iterative scheme for solving (9). Finally, we note that e,, c~ , c^ , o, of (3), (4), (5) and (6) are positive constants, independent of the dimension of M. In what follows, C will denote a generic positive constant, not necessarily the same in any two places.
264 W. N. Jureidini 2. Proof of Theorem 1. For a > 0 , consider
at f a t | u ( t ) | 2 = ^ a | , ( t ) | 2
+ A | M ( t ) | 2
<eat a C l | | u ( t ) | | 2+ | - | u ( t ) | 2 . (20)
Letting v = u(t) in (1), and using (4), (8) and the arithmetic--geometric mean inequality, we get
Jl |w(t)| = 2[ut,u]
= - 2 v (u9u)) + 2f,u)
<-2c2v\\u\\2 +2\\f\\J\u\\
< - * 2 ν | | Μ | | 2 + ( ν ^ Γ 1 l l / l l 2 . (21)
Using (21) i n (20) we o b t a i n
£eat \u(t)\2 < eat ( α β ι - ν β 2 ) \\u\\2 + eat (^> ^Γ1 l l f (* ) l l* · (22)
Now, i f a i s chosen so t h a t a e - v c 2 < 0 , we get
£ ( e a t \\u(t)\\Z) <eat\\f(t)\\l. (23)
I n t e g r a t i n g (23) over t h e i n t e r v a l [ 0 , T ] , we o b t a i n
| u ( T ) | 2 < e - a t | u ( 0 ) | 2+ i ^ | | | / | | | 2 ,
which imp l i e s ( 1 4 ) .
Next, l e t w(t) = u(t)—U. S u b t r a c t i n g (11) from (1) we get
[wt,v]+ ((w, v)) + b(u9u9v) -bU9U9v) =(f-F,v) , V y € M . (24)
A simple computat ion y i e l d s b(u 9u 9v) — b[U 9U ,v) = b (w 9U 9v) + b(u ,W ,v) ,
which when used in ( 2 4 ) , w i th V =w(t) y i e l d s
[w ,w] + v((w9w)) + b(w9U9w) = (f-F9w) . (25)
Thus,
_d_ at £ \w(t)\2 = 2[w,9w]
= ~2v((w9w))-2b (w9U9w) 4- 2 (f-F ,w)
<-2vc2\\wf + 2cJ\u\\ \\w\\+2\\f-F\\J\w\\
< " ( ^ - 3 ) | μ | | 2 + 3 " 1 l l ^ - ^ M j (26)
for any 3> 0, where we have used (25), (6), (8), (12), (13) and the arithmetic-geometric mean inequality.
Since λ > 0, we can choose ß= X/vc?1 in (26) and get
The Evolution Nav ier -S tokes Equat ions 265
£ ( e a t \\»(t)\\2) = eat(a\w(t)\2 + £ | w ( t ) | 2 )
< ea * ( « C l | | w ( t ) | | 2
+ i | | w ( t ) | 2 )
< ea t ( a C l - ^ A _ ) \\w(t)\\2
+eatve1 λ"ΐ | | / ( t ) - F | | 2 . (27)
Choosing a = s of Theorem 1, a c . - Ä / v c , < 0. Then i n t e g r a t i n g (27) over
[ 0 , 7 ] we get
M * ) | 2 < v ^ x-1 J eat~T) \\ft)-F\\\ dt + ,"αΤ μ ( ο ) | 2 , 0
which implies (16), since (15) holds. 3. Proof of Theorem 2. For a > 0 , consider
~, amk \ m\2\ amk, \ m+i \2 _. m\2\ 3(e I I ) = e y\u | + 3 \u l\ )
< e (ycl \\u II + 3 |w | ) , (28) where
γ = (eak - \ ) / k . (29)
Since [du ,u —u J < 0 , we have
Λ | / / I I 2 r ~ 777 777+1 , 7?Π
o\U \ = [d U , U +U
< 2 [ 3 M m , u m + 1 ] . (30)
Letting V = u in (9), and employing (4), (6) and the arithmetic--geometric mean inequality, we get
0 Γ~ m 777+ln _ , , 777+1 777+Kv . 0 / ^777 7 7 7 + l \ 2 [3 u , u J = - 2 v ((w , w )) + 2 ( / , u )
< - 2 ' v ^ i ^ ^ i l ' + zii^i i j i^+MI M
m + 1 | | 2 + J _ |iV"i|2
Making use of (30) and (31) in ( 2 8 ) , we obta in
< - v Ö 2 | | u m + 1 | | 2+ - i r l i m 2 · (3D
3(e w | ) < e (γ c - v c j u || + \\f || . (32)
But, from (29), γ -> 0 as ot->0. Hence, by choosing a sufficiently small, we can make γ c —vc2< 0, which from (32) yields
7 n amk d(e \u | ) < — — | | f II . (33)
Vi?2 *
From (10) and Schwarz i n e q u a l i t y we have
266
Λ 7 9 ° ^ &
W. N. Jureidini
(777+1 ) f c
< e am/c II f
mk (m+l)k
m^dt mk
(m+i)k
< ( eat\\f(t)\\ldt. mk
Using (34) in (33) we get
amk ( 7 7 7 + 1 ) ^
fe3(eu'"* \um\ ) < eat \\f(t)\\2 dt. •k z mk
Summing (35) for 777 = 0 , . . . , L — 1 , we o b t a i n
Lk aLk I L\2 i 0.2
\u \ - u < - i - J eat\\f(t)\\ldt 2 0
aLk . <- — Ulf III2
(34)
(35)
which implies (17). Next we establish (19). For a > 0 , we have
~ , α 777 k I 777 | 2 N amk , ι 77Z+1 |2 r m 7 7 7 - 1 - 1 777 η λ
3 ( e |ω I ) = e ( γ | ω | + |3u ,w ,w J) . a?77/c / || 7 7 7 + l | i 2
0 r ~ 77? 777+ 1 η χ ,~r. <e [y o \\w II +2\dw ,w J ) , (36)
where we have used the i n e q u a l i t y [dw ,W ] < [ 3 u ,W ] , which fo l lows
e a s i l y from the i n e q u a l i t y [dwmiwm+1 -wm] > 0 .
Summing (36) for 777 = 0 , 1 , . . . , L— 1 , we get L - l L-l
aLk\ L\2 1 0 i 2 ^ r-« 01777/ 11 777+1 | | 2 . v"1 o a w Γ^ W 777+ 1η e \w I - \w I < \ e Ύσ ||w || + \ 2e [3 w w J
777=0 L 777=0
777+1
(37)
Subtracting (11) from (9) and letting v = W * , we have
[ί/,/+1]=-ν((/+1,/+1))-Μ/,ί,/Η)+ (fm-F,wm+1) < - Y C 2 l l « m + 1 l l 2
+ C J | i / | | l l ^ l l lk m + 1 ! l +\\fm-F\\Jw
m+1\\. (38)
Using (38) in (35), the inequality L-l L-l
0 \ a7??k | | 777 | | | | 777 + 1 u e i ' w 'I I ' w
»7-1-1 it ii ON . / , . ak\ sr amk w 77?+iM2
II < \\w II + (1 +e ) ) e \\w || , 777=0 777= 0
and the arithmetic-geometric mean inequality we get
T h e Evo lu t ion N a v i e r - S t o k e s E q u a t i o n s 267
L-l JLLkt L i 2 , 0 i 2 e aLk\ Lxi I 0.2 ^ V amk I ^ γ - λ \ m+i„2
? \w I - \w | < / fee γ + II w ||
m=0 z
+ c Y ^ a ^ | | / " - F | | 2+ c | | « ° | | 2 . (39)
777=0
Choosing a = s , then from ( 1 8 ) ,
γ + — < 0 . (40)
Moreover, by an argument similar to that in (34), we have £ - i Lk J ke«mk\\f-F\\ < \ eat\\f(t)-F\\2 dt. (41)
m=0 0
Using (40) and (41) in ( 39 ) , we o b t a i n
aLk I L.2 i 0.2 ^ _ ,, o ii 2 e I M — U M < C W + £ f eat\\f(t) ~F\\l dt
0 which y i e l d s
Lk
\wL\2 * e~Lk(\wu\2 +C\\J\\2) +C [ e8^-Lk)\\f(t)-F\\Jt.
0 Taking the limit as L -* °° , and using (15), we get (19).
REFERENCES
1. W.N. Jureidini, A Mathematical Study of Two Dimensional Incompressible Viscous Flows, Ph.D. Thesis, Harvard University, 1980.
2. J.L. Lions, Quelques Methode de Resolution des Problemes aux Limites Bonlineares, Dunod, Paris, 1969.
3. L.E. Payne, On the stability of solutions of the Navier-Stokes equations and convergence to steady state, SI AM J. Appl. Math. 15 (1967) 392-405.
4. R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Elsevier, North Holland, 1977.
ITERATIONS D'ALGORITHMES D'ACCELERATION DE LA CONVERGENCE
M. EL DJALIL Departement de Mathematiques, Infermatique
Universite d'Oran Es-Senta, Algerie
A b s t r a c t
Logarithmic sequences are usually difficult to accelerate. In this paper, we show how to accelerate these sequences by the interation of classical algorithms. We give theoretical results, and illustrate the results by numerical examples.
RESUME
Les suites ä convergence logarithmique sont souvent difficiles a accelerer. Nous montrons comment accelerer la convergence de ces suites par l'iteration de procedes classiques. Des resultats theoriques sont donnes et concretises par des exemples numeriques,
Introduct ion
Soit (S ) une suite de nombres reels de limite S. Toutes les n transformations de suites pour accelerer la convergence consistent a transformer la suite (S ) en une suite (S ) verifiant:
n n s( 1 ) - s
lim "s - s °-m->°° n
269
270 M. El Djalil De t e l l e s t r a n s f o r m a t i o n s s ' e c r i v a n t : ,
( 1 ) T : (S ) > ( S ( 1 ) = S - -ττ^- · A S ) n n n Ab n
n oii (b ) est une suite de limite nulle construite a partir S , n n S ,, , S ; ρ etant un entier positif non nul. Dans n+1' ' n+p v ^ la pratique en remarque que 1'iteration de procedee T definis comme en (1) avec un choix de la suite (b ) , accelerent la con-
n vergence de certaines suites a convergence 1ogarithmique. Le but de ce papier est de justifier ces remarques theoriquement et de les illustrer par quelques exemples numeriques.
1 Notations et Definitions
Definition (1.1) [1]: On dit qu'une suite (S ) est a convergence n
logarithmique si: Sn+1 " S
lim — — — = 1 ; S designant la limite de la suite n-*» n
convergente (S ) .
Definit ion (1.2) [1]: On dit que la suite (S ) appartient ä LOGSI si:
AS lim " = 1 : oü AS = S . - S AS n n+1 n n-*°° n
Definit ion (1.3) [1]: Soit T le procede transformant la suite (S ) en la suite (S ) . On appelle iteration de T la procedure
n v n (1) (2) -(2) consistant ä transformer (S ) par T en (S ) puis (S ) puis
(2) (3) n n n (S^Z;) en (SKO)) etc n n
Def init ion (1.4) [6]: Un ensemble E de suites convergentes est dit stable par un precede T si pour toute suite (S ) de E la suite (T(S )) appartient aussi ä E. n
Definition (1.5) [6]: On dit qu'une suite (u ) de limite nulle n
verifie la propriete A1 si ä partir d ' un certain rang on a:
Iterations d'Algorithmes d'Acceleration de la Convergence 271
u . = f ( u ) oii f e s t u n e f o n c t i o n a n a l y t i q u e au v o i s i n a g e de n+1 n z e r o e t t e i l e q u e :
f ' ( 0 ) = 1 ; f' ' ( 0 ) = . . . = f ( k ~ 1 ) ( 0 ) = ( 0 ) e t f ( k ) ( 0 ) * 0 a v e c k C N*
Def init ion (1.6) [6]: Soit (t ) une suite convergente de limite nulle. On dit que (t ) verifie la propriete Bf s'il existe un entier m non nul et une suite (u ) de limite nulle verifiant la
n propriete AT et des constantes ( α . ) . > π tels que:
1) a° = 0 si k < m ; a° * 0 k m
o\ o m o m+1 , o m+i 2) t = a° u + a° „ u + . .. + a , . u + . . .
n m n m+1 n m+i n pour n suffisamment grand.
2. Presentation du Probleme
Dans [3] GERMAIN-BONNE ET DELAHAYE ont montre que LOGSF n'est pas accelerable.
Nous nous proposons alors de caracteriser un sous-ensemble strict de LOGSF qui soit accelerable par l'iteration de certains procedes classiques definis comme en (1).
Notre idee a consiste ä chercher un sous-ensemble de LOGSF que 1'on notera par SLOG tel qu'on ait pour certains choix de la suite (b ) : n
- La stabilite de SLOG par T - L ' accerabilite de SLOG par T.
3. Resultats Theoriques
La caracterisat ion de SLOG repose sur la proposition suivante:
272 M. El Djalil
PROPOSITION: S o i t (S ) une s u i t e c o n v e r g e n t e de l i m i t e S e t n
verifiant l'hypothese H suivante: AS n+1 AS 1 + λ ou (λ ) est une suite de limite
n n' AS
(H^
nulle teile que lim -=— = 0 (regularite du δ2 d*Aitken n->oo n
pour la suite (S ))
et (λ ) verifie la propriete B1 avec plus precisement: n
1 im
1 im
»»♦1 " % . f(m+1)(0) „ 0 e t m+ 1 (m+1)!
r(m+1) n->°° u (m+1)! n
(C) * 0; m etant un entier ^ 1
On suppose (b ) de limite nulle et teile que:
(H2) <
n+1 1 + g oü (g ) est une suite de limite nulle n n
s'ecrivant g =g(X , λ „, ..., λ „); p etant un n n n+1 n+p-1 entier non nul. On suppose de plus que (g ) verifie 1ε
propriete B' avec:
lim — = (1 im — ) -n+°° u n-+°° u (m+1 ) I
_m f ( m + D ( 0 )
PREUVE:
Sous ces hypotheses la suite (S ) converge vers S et verifie: (1> S ( 1 ) - S
1) lim — ^ = lim τ^ — = 0 AS AS S - S
n->oo n n->°° n AS (1)
2) j-r*- = 1 + λ ; oil λ est une suite de limite A s(1) n n
nulle verifiant la propriete Bf.
Iterations d'Algorithmes d'Acceleration de la Convergence 273
De p l u s q u a n d (λ ) e t (g ) v e r i f i a n t l a p r o p r i e t e Bf a v e c m = 1 ,
AS 1 im n->°° λ
(1 ) (1 )
( 1 ) 0 e t l i m
n-*oo u m ( m + 1 ) !
m f ( m + 1 ) ( 0 ) * 0 .
4. Applicat ions
a) Exemples de suites (S ) verifiant (H„) quand m=1 n 1
1) Soit (S ) une suite convergeant vers S et teile que:
/oo
e d(|)(t) ; ou (J)(t) est une fonction 0
verifiant les proprietes (H.) suivantes:
(ΗΑ) :| φ ( 0 est non decroissante sur [0,°°[ φ ( 0 est entiere sur [0,+°°[ ΙφίΟ I ύ M, V t e 1R
lim (logUT (t) L < Itl
Widder mentre dans [7] qu'une teile suite s'ecrit:
s n - s = ^ + V i +...+ V i + . p p +1 p +1 nr nr nr P £ 1
et ce pour n assez grand; on en deduit alors facilement que (S ) verifie (H„) quand m=1. n 1
Les exemples rencontres dans la litterature sont nombreux; on retiendra surtout:
1 . <h(t) = - e et S oo
-/· e " ( n + 1 n J Λ dt = 1
(n+1)!
) ( t ) = - t e t - e t e t S n " (η+1)κ
3. On de*finit φ ( 0 par φ!(ϋ) = ß(t) = (£ —^ ) e~2 t
e - 1
274 M. El Djalil
/°o n+1
e d t = Σ T - L o g ( n + 1 ) > γ 0 m=0 m + 1
oü γ est la constante d'Euler.
2) Soit S une suite de limite S = Σ a en suppose n . v. A n n^O
n que S = Σ a, avec a = / e d(b(t) ou φ verifie (H.) et n k=Q k n JQ φ OO
l ' h y p o t h e s e s u p p 1 emen t a i r e : 1 im / e d(f)(t) = 0 . x->°° 0
On montre aisement que la suite (S ) verifie l'hypothese (H..) quand m=1.
3) Soit (S ) une suite de limite S teile que S = F(S ) n n+ Ί n ou F est une fonction de OR dans OR verifiant:
F est analytique dans un voisinage de S; S etant le seul point fixe dans cet Intervalle et F'(S)=1. S ainsi definie verifie
n alors l'hypothese (H.,). La demonstration est triviale.
Nous proposons de donner des exemples de procedes T accelerant ces types de suites; ces procedes etant determines par le choix de la suite (b ) , nous allons done donner des exemples de suites x r
(b ) verifiant l'hypothese (EL) avec m=1. Nous appelerons demormais SLOG I'ensemble des suites (S ) verifiant (H,) avec
n 1 m=1 .
b) Exemples de suites (b ) verifient (H0) avec m=1 n z AS
1) b -n AS j „ 1 - n + 1
AS
AS AS Λ n+1 Λ , r Comme 1 = —τ-ς = -λ en a b = r— AS n n λ n " n
b . AS . λ λ n+1 n+1 n , , , λ η et done 1 im b = 0 . De plus —r = —τ -ζ · τ = (1+λ )-ζ-n b AS λ . n A . n->°° n n n+1 n+1
Iterations d'Algorithmes d'Acceleration de la Convergence 275
On montre alors (b ) verifie ( Η ? ) .
2) b - n + 1 n AS A 1
1 n + 1 AS n
On montre comme pour l'exemple precedent que SLOG est stable et accelerable par le precede
T ! ( s ) > (s^>) = (sn+1- - ^ 5 _ . Δ8η+1) ii"<-1
T est en fait le θ9 algorithme ([1]). L'iteration de T donne (k) alors les suites (S ) verifiant: n
ii s ( k +° - s p ™> «> P = o pour tout k Z 1
s(k) - s n
3) b = n Δ S n — n
On montre que SLOG est stable et accelerable par le precede
T s (sn) > (Sn - ^ - · Δ 8 η ) . n
T est en fait la premiere colomne du u-algorithme de Levin [4] (k)
L'iteration de T donne alors des quantites (S ) verifiant:
lim S < k + 1 ) - S n >°° T—T = 0, pour tout k £ 0
s u ; - s n
5. Resultats Numeriques
Nous avons teste l'iteration des trois precedes du chapitre precedent et avons obtenu les resultats numeriques suivante;
Exemple 1: Soit (S ) la suite generee par:
9 et Sn = 2,5 n+1 6 - S "0
La suite (S ) est convergente vers 3.
276 M. El Djalil
Avec 1 ' i t e r a t i o n du p r e c e d e 1 nous avons ob t enu :
3 (2) 3 (3) 3
,298888888888889 D + 01
,299977135234993 D+01
.299999955518070 D+01 ,(3) Pour le calcul de S.: on a utilise S S ?, ..., S.,.
Avec l'iteration du precede 2, nous avons obtenu:
S^1^ = .300000000000000 D+01
Pour ce calcul nous avons utilise S., S ~ > ..., S,. titre de comparaison, on a:
S 5 0 = .294545454545454 D+0 1
Exemple 2: Soit (S ) la suite definie par:
S = Σ — n k=1 k2
La suite (S ) est convergente de limite S = —p-
Avec l'iteration du precede 1, nous avons obtenu:
3 (2) 3 (3) 3 (4) = 3 (5) 3
= .161521371312176 D+01
= .164384663154480 D+01
= .164491418652957 D+01
164483321713539 D+01
= . 164493489238477 D+01
Pour ce calcul, nous avons utilise S , ..., S
Avec l'iteration du precede 2, nous avons obtenu:
1 163888888888889 D+01
s j 2 ) = .164492258652055 D+01 ,(3) = . 164493405576917 D+01
Iterations d'Algorithmes d'Acceleration de la Convergence ς(4)
c(5) = .164493411508698 D + 01
= .164493406662454 D + 01
277
Ce calcul a necessite la connaissance de S,, .... S „ . 1 * ' 1 6
Avec l'iteration du procede 3, nous avons obtenu:
164505669664439 D+01
164492438593204 D+01 .(5) . 164493387128660 D + 01
Ce calcul a necessite la connaissance de S.., ..., S 24'
A titre de comparison, on a:
349
>50
162473273362152 D+01
162513273362152 D+01
Exemple 3 : SOIT la suite (S ) definie par:
3 = Σ 2, avec n . ^ k k = 0
2k =
F4-T + L O ^ Ü T T ) s i k*°
+ 1 si k = 0
La suite (S ) est convergente de limite C=0,577215664901533.
Avec l'iteration du procede 1, nous avons obtenu:
,593943246524111 D+00 .(2) Sj ' = .577814376638739 D+00
S^3) = .577226659305818 D+00
S^4) = .57721567595724 D+00
S^5 ) - .577215697845751 D+00
Ce calcul a necessite la connaissance de S , ..., S1 .
278 M. El Djalil Avec 1'iteration du procede 2, nous avons obtenu:
S^1) = .577403144179682 D+00
S^2 ) = .577220487295191 D+00
S^3^ = .5772156639468820 D+00
S^4) = .5772156663953863 D+00
S^5 ) = .577215664785223 D+00
Ce calcul a necessite la connaissance de S., ·.., S.0. i Ί o
Avec 1'iteration du procede 3, nous avons obtenu la meilleure approximation avec:
S ^ = .577215764015521 D + 00
Ce calcul a necessite la connaissance de S,, ..., S O Q .
A titre de comparaison, on a:
S / Q = .587385040218799 D+00 49 S5() = .587182332901279 D+00
Nous remarquons a travers ces exemples que 1fen a acceleration de la convergence des suites (S ) par l'iteration des procedes 1 , 2 et 3.
Nous tenons ä faire remarquer que d'autres methodes ont ete e*laborees dans le cadre de 1'acceleration de suites ä convergence logarithmique; nous citons en particulier le travail de BREZINSKI-DELAHAYE-GERMAIN-BONNE [8], qui consiste a extraire de la suite originale une sous-suite a convergence lineaire qu'on arrive done ä accelerer par les procedes classiques tels le 62 d'AITKEN. Nous remarquons cependant que l'algorithme d'extraction s'avere assez couteux; en effet prenons l'exemple 2 qui consiste ä estimer , Σ. 1/k2. Nous arrivons avec les 16 premiers termes de la suite a un resultat meilleur qui celui obtenu avec l'algorithme d'extraction [8], qui, lui, utilise les 245 premiers termes de la suite.
Iterations d'Algorithmes d'Acceleration de la Convergence 279 BIBLIOGRAPHIE
1. C. Bresinski, Acceleration de la Convergence en Analyse Numerique. Springer-Verlag (1977).
2. C. Bresinski. Algorithmes de 1'Accel erat ion de la Convergence. Etude numerique. Technips (1978).
33 J. P. Delahaye et B. Germain-Bonne, The set of logarithmically convergence sequences cannot be accelerated, SIAM J. Numer.
Anal. (1982), pp. 840-844. 4. D. Levin, Development of non linear transformations for
improving sequences, Internet. J. Corny. Math. B5 (19 ) , 371-388.
5. P. Wynn, Transformations desseries a 1'aide de 1'algorithme. C.R.A.S. A 27E (1975) pp. 1351-1363.
6. M.E.D. Kageb, Iterations d'algorithmes d'ace el eration de la convergence. These de 3eme cycle, Lille (1983).
7. D. Widder, The Laplace Transform. Princeton University Press (1946).
8. C. Brezinski, J. F. Delahaye and B. Germain-Bonne, Convergence acceleration by extraction of linear subsequences. S.I.A.M. Num. Anal. 20 (1983) No 6, 1099-1107.
A THIRD ORDER FINITE-DIFFERENCE METHOD FOR TWO DIMENSIONAL
PARABOLIC EQUATIONS
A. Q. M. KHALIQ and A. Y. AL-HAWAJ Department of Mathematics, University College of Bahrain,
P.O. Box 1082, Bahrain
1. Introduction. Parabolic partial differential equations represent a class of problems of great importance in many fields of engineering and science. When the need for a numerical solution of such problems arises, the requirements for computing time and storage capacity often tax the capabilities of today's largest digital computer. Thus the need of more efficient numerical methods is an ever constant one.
When solving numerically a two dimensional parabolic partial differential equation it is always advantageous, if not desirable, to use splitting methods (see, for example, [2]). The origin of the use of splitting methods for parabolic equations can be traced to the pioneering work of Peaceman and Rachford [6] and Douglas [1]. Various other splitting methods have been discussed in Gourlay [2].
In a recent paper [4] it was shown that Peaceman-Rachford method gives unsatisfactory results for solving two dimensional parabolic equations in which high frequency components occur in the solution when a time discretization is imposed with time steps which are too large relative to spatial discretization. Lawson and Morris [4] developed a second order L -stable method as an extrapolation of low order method based on the (1,0) Pade' approximation to the exponential function which rapidly damps high frequency components. Khaliq and Twizell [3] achieved third order accuracy as an extrapolation of second order method based on the (2,0) PadeT approximation to the exponential function which retains the property of L -stability. The essential theme of each of these papers was to develop LQ-stable method for the numerical solution of parabolic equations with non-smooth initial data.
281
282 A. Q. M. Khaliq and A. Y. Al-Hawaj The present approach differs in the manner in which the increased
accuracy is achieved. In the present paper it is assumed that the coefficient of diffusion is constant. If it depends on the space variables (s),no change to the given algorithm is necessary. The extension to the present work is the consideration of time-dependent coefficients and equations with inhomo-geneous source terms which will be considered in a future paper.
In Section 2 the difference scheme for the solution of two dimensional parabolic equation is discussed. Section 3 represents the results of numerical experiments and their explanations using the method outlined in Section 2. The paper is concluded in Section 4.
2. Third Order LQ-Stable Method. Consider the constant coefficient parabolic equation in two space variables given by
2 2 -rr = —2 + Γ Τ > 0 <x9y <X9t> 0 (1) dt dxl 82/
with homogeneous Dirichlet boundary conditions on the boundary 3Ω of the
square Ω defined by the lines x = 0, z/ = 0, x = X, y = X, and in i t i a l condi
tions.
ux,y,0) = g ( x 9 y ) . (2)
Further it is assumed that g(x,y) does not necessarily have the value zero for (x, y) £ 3Ω, so that discontinuities between the initial conditions and boundary conditions are permitted.
Both intervals 0 < x, y < X are divided into N+\ sub-intervals each of width h, so that (N+ 1 ) h = X . At each time level t = nl (n=0,l,2,...) the square Ω together with its boundary 8Ω have been superimposed by a
2 square mesh with N points within Ω and N+2 equally spaced points along each side of 3Ω .
The solution u(x >y ,t) of (1) is sought at each point (kh, mh , nl) in ΩΧ [t>0] where k,m = 1,2,...,# and n=0,l,2,... . Denoting the theoretical solution of an approximating difference scheme at the mesh point (kh , mh, nl) by U-, , ' J k,m the form
u(t) - nn- (jjn jjn jjn . rjn jjn nn .
• • • ; U",N ' U2,N " · " UN,Nj ' ( 3 )
where T denotes transpose. The space derivatives in (1) are now replaced by second order finite
difference approximations 2
- γ τ = h~2 ux-h,y,t) -2u (x,y9t) +ux+h,y,t) + 0(/z2) , (4) 3 x
Two Dimensional Parabolic Equations
d2u
283
(5) -2-2 - h 2u(x,y-h9t) - 2u (x,y,t) + ux,y+h9t) + 0(h2)
at each time level t = nl (n = 0, 1,2,...), (1) is applied to all N2 interior grid points of the square Ω with the space derivatives replaced by (4), (5).
2 These N applications result ential equations of the form
2 . 2 These N applications result in a system of N first order ordinary differ-
dU(t) ~dt~ = A U(t), (6)
2 2 The matrix A in (6) is N X N and may be split with the constituent matrices B, C such that A = B+ C. It is assumed that the matrices B and C do not commute (see, for example, [7], [8]). Hence equation (6) becomes
dU(t dt = (B + C) U(t) . (7)
In (7) the matrix B is of order N and arises from the use of (4) in (1); it is block diagonal with tridiagonal blocks and has the form
1-. 0 *Bi
(8)
where 5 is the tridiagonal matrix of order N given by
-2 1
Bl =
1 -2 L
-2
(9)
The matrix C is also of order N and arises from the use of (5) in (1); it is block tridiagonal with diagonal blocks and has the form
C = h~2
21
I >»
I -21
v
0
V.
I s X
"I
s
0
*v
-21 I
X I -21
(10)
where J is the identity matrix of order N.
Solving the system of ordinary differential equations (7) subject to the initial vector U(0)=g gives
U[t) = exp [tB + C)]g
which satisfies the recurrence relation
284 A. Q. M. Khaliq and A. Y. Al-Hawaj
U(t+ I) = exp[Z(£ + C)] U[t) , £ = 0 , Z , 2 £ , . . . , (11)
which may be approximated using rational approximations to the matrix exponential function.
The following splitting of (11) is proposed using the (2,1) Pade' approximation to the matrix exponential function in (11) to achieve third order accuracy.
U* = (l~lB + - B2\l ΓΓ + I B V J + J Z C +-VC'2)~1 \I + \c\ut) (12)
( j - ^ ^ + z V ) " 1 ( l+ | z s ) ( l - | ^ + ^ C 2 ) " 1 ( l + J c ) ^ * ) . (13)
Expanding the matrix inverses in (12) and (13) it is seen that neither (12) nor (13) is third order accurate when compared with the Maclaurin expansion of exp[Z(5 + C)] in (11) which gives
U(t+t) =[l+l(B + C) + | I2 B2 + C2 +BC + CB)
+ | z 3 ( 5 3 +C3 +B2C + BC2 + C2B + CB2 +BCB +CBC) + ...]ü(t) . (14)
However, a l inear combination of (12) and (13) defined by
U(t + l) = (§ ί / + - | ί / * ) + ο α 4 ) (15)
is seen to be third order accurate when compared with (14).
Stability of (15) is analysed by writing (11) in the form
U(t+l) = exp(U) U(t) (16)
and replacing the matrix exponential function by its (2,1) Pade1 approxi-mant, which gives
I + \A U(t + l)= — — U(t) + 0(Z4). (17)
I-\lA + \ I2 A2
The amplification factor of (17) is given by
s2>1(*) = (i-|2)/(i+|2 + | 3 2 ) ,
where z = - l \ , λ an eigenvalue of A. For negative definite A, 2 > 0 and
lim S2 γ[ζ) = 0
which proves that (15) is L -stable. The principal part of the local truncation error of (15) at the
interior mesh points (kh , mh , nl), (m = 2 ,...,/!/- 1 ; n = 0 , 1 , . . . ), is given by
Two Dimensional Parabolic Equations 285 1 I,1 d*u\ _1_ ,3 _8 u
^ 3yV 72 ^ J m . f e For computational purposes the relation (15) is formulated by the
following algorithm which requires six applications of quintdiagonal solvers.
j _ 2 ^ + ^ T / ( l ) 7 0 ) = (i + \ c ) U(t) ,
(18)
U(t+l) = ±(9U -ί/*)+0(ΖΗ) .
In (18) 7^^,7^2\ t^3),^1*) are intermediate vectors each of order N2.
In practice these vectors can be over written to save the storage. The quit-diagonal coefficient matrices involved in (18) may be decomposed in LU
decomposition to solve the system of linear equations.
3. Numerical Results. To examine the behaviour of the novel method discussed in Section 2, it was tested on two problems from the literature as follows.
PROBLEM 3.1, [4]. Consider the equation
du dzu d^u 2t dx2 dy2 0 < x,y < 2, t> 0
subject to the initial conditions
ux>y90) = s i n ( - 7 r z / ) ; 0 < x , y < 2
and boundary c o n d i t i o n s
u(x,y9t)= 0 ; x = 0, z/ = 0, x = 2, y = 2, £ > 0
The t h e o r e t i c a l s o l u t i o n i s g iven by
ux9y,t) s i n 2 "ny [ l - ( - 0 K ] ~ s i n ( i K i r x ) e x p ( - l n 2 ( K 2 + i ;
286 A. Q. M. Khaliq and A. Y. Al-Hawaj The maximum value of u at time t= 1 .0 occurs for x = 1 , y = 1 and is approximately 0.01. Following Lawson and Morris [4], solution is computed using fixed time steps 1 = 0.1 and 1 = 0.01. The maximum errors in the computed solution for this experiment using the Lawson and Morris [4] LQ-stable method, the Peaceman-Rachford [6] 4Q-stable method and the novel method (18) are included in Table 1. The maximum errors for the Peaceman-Rachford method occur near the boundaries z/^O, y - 2 when the time step 1 = 0.1 is used with space steps of h=0.Q5 and h- 0.025. This shows the unsatisfactory behaviour of Peaceman-Rachford method when time steps are used which are too large relative to the space discretization. The numerical solution exhibits an oscillatory behaviour which increases in amplitude with a reduction in spatial discretization, keeping the time step constant. In contrast, the Lawson-Morris L -stable method and the novel method (18) do not suffer from these drawbacks. The maximum error for the novel method (18) occurs at the point where x- 1 , y = 1 . Visual analysis of Table 1 shows that Peaceman-Rachford method gives good results when the ratio r=l/h is not too large. The novel method (18) is seen to give superior results to Lawson-Morris method and Peaceman-Rachford method. The novel method (18) gives higher accuracy to those of [4] and [6] at the expense of an increase in CPU time. However, superior results justify this minimal increase in computer time.
PROBLEM 3.2, [5]. Consider the equation
du d U , du n ^ ^ , , ^ n dt dx2 dy2
sub jec t t o i n i t i a l c o n d i t i o n s
u(x 9y ,0) = sinTr;c sinfri/ ; 0<x,y< 1
and the boundary c o n d i t i o n s
2u = π exp (-2 π 2 t) s i n i\y ; x = 0 1 dx \
^ + 2u = - π ε χ ρ ( - 2 i r 2 t ) sim\y ; ar= 1 J ax
2u = π ε χ ρ ( - 2 π t) s i n i rx ; 2/ = 0 i
— + 2u — — π εχρ ( - 2 π t ) s in i rx ; y = 1 J du
0 < y < 1 , t > 0,
0 <x < 1 , t > 0.
The theoretical solution to the problem is given by
u(x ,y >t) = exp (-2π t) sin π# sin n\y .
The solution is computed using the Peaceman-Rachford method and novel method (18) taking /z=0.05 and 1 = 0.0025. The maximum error (E), for both
Two Dimensional Parabolic Equations 287 methods at a grid point after a number of time steps (P) is shown in Table 2. The novel algorithm appears to perform best in that the maximum error appears to be much smaller than that of Peaceman-Rachford method.
Table 1. Maximum Errors in Solving the Problem 3.1 at t = l
Method
Lawson-Morris Peaceman-Rachford Novel scheme
Z = 0.1 h = 0.05 1 /2 = 0.025
0.88(-3) 0.23(-l) 0.12(-4)
0.87(-3) 0.46(-l) 0.91(-5)
Z = 0.01 /z=0.05 | /z = 0.025
0.35(-4) 0.16 (- 4) 0.28(-5)
0.22(-4) 0.55(-5) 0.17(-5)
Table 2. Maximum Modulus Error in Solving the Problem 3. 2 at a number of time steeps (P)
P
10 20 40 100
Modulus Error Peaceman-Rachford
Method
0.795(-3) 0.456(-3) 0.198(-3) 0.431 (-4)
Novel Scheme
0.331 (-3) 0.187(-3) 0.973(-4) 0.125(-4)
4. Discussion. The paper deals with the numerical solution of parabolic equation in two space variables. The time dependent partial differential equation is transformed into system of ordinary differential equations before computing the solution and third order splitting method based on the (2,1) Pade' approximation to the matrix exponential function has been developed. The novel method is L -stable, removes constraints on the discrete time steps, and rapidly damps high frequency components, which is of importance for problems have discontinuity between the initial and boundary conditions. The method also behaves well for problems with smooth initial data having derivative boundary conditions.
REFERENCES 1. J. Douglas, Jr., On the numerical integration of (9 u)lx +(9 u)fdy< = (du)It
by implicit methods, STAM J. Numer. Anal. 3 (1955) 42-65. 2. A.R. Gourlay, Splitting methods for time dependent partial differential
equations in The State of the Arts in Numerical Analysis, (Ed. D.A.H. Jacobs) Academic Press, London, 1977.
3. A.Q.M. Khaliq and E.H. Twizell, LQ-stable methods for parabolic partial differential equations, Brunei University Department of Mathematics and Statistics, Technical Report TR/02/82, 1982.
288 A. Q. M. Khaliq and A. Y. Al-Hawaj
4. J.D. Lawson and J.LI. Morris, The extrapolation of first order methods for parabolic partial differential equations I, SI AM J. Numer. Anal, 15 (1978), 1212-1224.
5. A.R. Mitchell and D.F. Griffiths, The Finite Difference Methods in Partial Differential Equations, John Wiley, Chichester, 1980.
6. W.D. Peaceman and H.H. Rachford, Numerical solution of parabolic and elliptic differential equations, SIAM J. Numer. Anal. 3 (1955), 28-41.
7. E.H. Twizell, Computational Methods for Partial Differential Equations, Ellis Harwood Ltd, Chichester, 1984.
8. E.H. Twizell and A.Q.M. Khaliq, LQ-Stable Methods for Parabolic Partial Differential Equations, Brunei University Department of Mathematics and Statistics Technical Report, TR/02/82, 1982 (Revised).
A PARALLEL ASYNCHRONOUS RELAXATION ALGORITHM FOR OPTIMAL
CONTROL PROBLEMS
J. C. MIELLOU and P. SPITERI U.A. CNRS No. 741, LANI, Route de Gray,
25030 Besancon Qedex, France
Abstract. This paper deals with the formulation in a mathematical way of a parallel asynchronous relaxation algorithm intended to resolve optimal control problems; furthermore it yields convergence results in a general framework. Technical applications involve the control of thermal processes for which results of numerical simulations are given.
1. Introduction. In order to control complex continuous plants on line and to react without delay to any perturbation, it can be necessary to solve a two point boundary value differential problem, derived from Hamilton-Pontryagin optimality conditions very quickly; taking into account the current development of computation techniques, our interest mainly lies in the determination of a control optimal law using relaxation algorithms which match parallel synchronous or asynchronous computers particularly well. The chief advantage of this method is indeed to reduce computation time appreciably and to allow interconnected processes to be controlled on line.
In this paper we propose one parallel asynchronous variant of the costate relaxation method [12] in which parallel processes send available computation in a reciprocal way, by exchanging sets of values, each set corresponding to the calculation obtained in a subset of the horizon; thus, owing to asynchronous communications between the processes, the interaction functions are piecewise continuous. The algorithm can be defined in a mathematical way and its behaviour can be analysed using some adapted notion of accretivity [1] which shows that the fixed point mapping related to the problem is contracting for an adapted scalar norm [14] in the set of piecewise continuous functions; by applying a result of El—Tarazi [7] the previous MAA—T
289
290 J. C. Miellou and P. Spiteri property guarantees the convergence of the asynchronous iterations. Note that the convergence criterion which is obtained is independent of the size of the horizon and is applicable in the case where control is subjected to some constraints; on the other hand, this method is applicable whether or not the objective is a function of the terminal state. Other extensions to more general problems can be considered for example, singular perturbation problems in control theory (which correspond to systems with slow and fast modes) or control problems with several cost functions (see [8]).
This theory is applied to the study of the regulation of a thermal process for which results of numerical simulations are given (see [17]).
2. General Formulation of the Problem. Let n and m be two integers, U
the space of the bounded and piecewise continuous functions on [t , T~\ which has values in if: let U j be a closed convex subset of U, which is the domain of admissible controls. Furthermore, suppose that the state of a process is described in the n-dimensional space, by the following differential equations:
y(tQ) = o ·
vt e [tn ,τ] (i)
We want to drive this system from t to T> starting from the initial state, and minimizing the objective function:
T J = J ry9u9t) dt (2)
under the supplementary c o n s t r a i n t
u(t) £U ad * v t e [ t 0 , T ] . (3)
We suppose that / and v are continuous functions of y and u ; in addition, we assume that / and r have continuous partial derivatives with respect to y and u.
When the terminal state is free necessary optimal conditions are given by the Hamilton-Pontryagin minimum principle [15] as follows:
dt
y(tQ) = o,
dt \dyJ
v t e [ * 0 , T ]
_9P 'by o , v t e [ t n , T ] (4)
p(T) o,
U \du J P+ a ψ B O , Uad
A Parallel Asynchronous Relaxation Algorithm 291 where p(t) is the costate, A denotes the transpose of the matrix A, and 3 ψ^ is the subdifferential mapping of the indicator function ψτ, for
ad uad the convex set Ua(^,
3. Sequential Algorithm: Costate Relaxation Method. We recall now the so called costate relaxation algorithm [12] which consists in alternatively integrating, in sequences, state equations and costate equations, respectively in direct and reverse sense, then in solving the remaining inequalities. This leads to the following iterative operations: (i) initially, choice of a first approximation for optimal control law, written u^°^(t),
(ii) determination of the state variable y ^ (t) , where q is the number of the iteration, by integration of the state equations:
^ / b ( i t i ) . » ( , ) . ^ » 1 dt f (5)
(iii) determination of the costate variable p " (t) by integration, in the reverse sense, of the costate equations :
- ^ ^ ^ P ( i + 1 ) - ^ + l).«W.*)-0 , (6)
^+1)(T) = 0 where
! —*- I and a l υ . u: . t 1 = -— KTy) and *<*·"»*> " 3 7 ' (iv) resolution of the inequalities
h ( y ^ + l ) , u ^ + 1 \ t ) + C p ^ + l )+ ^ n (u^+l))30
ad (7)
C=-(l£-Y and h(y9u,t)= |^ . This methodology corresponds to a Gauss-Seidel strategy; Jacobi's
strategy should be obtained by replacing y ^ by y ^ in (5), (6), (7), and p(< ? + l ) by p ( ^ in (6), (7).
4. Parallel Asynchronous Algorithm : Costate Relaxation Method with Asynchronous Communication. The asynchronous parallel approach consists in simultaneously integrating state equations and costate equations, using two processors; each processor sends the results of its computations to the other in the form of component packages; the packages correspond in fact to a set of values on an interval Γτ, , τ7 , left τ] . The control law is determined
L h h+lJ 0' J
indifferently by either processor, eventually by a third. Let us specify
292 J. C. Miellou and P. Spiteri that the asynchronism is due to the fact that each processor operates individually at its own pace using the latest available data (interactions).
This algorithm can be formally modelled in a mathematical way; first, we consider the following notations:
Let a and V be two integers with a = 3. Consider now a set of real numbers τ1 ,» . . , τ such that
v (Th+l~Th) =T-tQf with T l = * 0 and τν = Τ
->n \
and define the following spaces :
*i,Ä = C < ] V r A + 1 ] ; Ä " .
where C ) denotes the set of continuous funcions, and let:
(8)
with
and
7
E
X
—
=
II 7z=i
a
(χ^
E .
E .
% yi
h '
5 3
i = 1 , . . . ,α
defined by :
1 dt
(9)
(10)
Ä1(i)=7(2/,M,t) ,
λ2Χ) = Αρ -g(y9u9t) ,
Λ3(ί)= Cp+ h(y,u9t) .
Finally, consider the set of diagonal operators Λ. for i € 1 , . . . , α
D(h,) being the set of continuous n-dimensional functions on [t , T] , piecewise C on [£. , T] with respect to the subsets Γτ7 , τ7 Γ and which
0 r L h n+lL vanishes when t — tn ,
A Parallel Asynchronous Relaxation Algorithm 293 Ζ)(Λ„) being the set of continuous n-dimensional functions on [t , T~\ , piece-wise C on [t , T~\ with respect to the subsets ]τ, ,τ, ] and which vanishes when t = T; and
A? = 3Ψί, O(A^) being Vad.
With the previous notation, system (4) can be written :
A^(X) +Λ ( Ϊ)30, Vt£ !,..., a.
We can now associate a fixed point equation to system (11) as follows:
3 £ + / ( l / . t > . * ) = 0 , V t 6 ] T A , T f t + 1 ] , V Ä = 1 v - 1
( I D
~ + Ap-g(z9v9t) = 0 , V t e [x f e , Tk+1[ , V f e = v - l , . . . ,
M^.Ujt) +^φ + 9ψτ/ (w)30,
(12)
ad where ι/(τ,) and p ( τ * ) are determined by the continuity of y and p in
Let f/ = ζ,φ,ν where s(t),ip(t) and u(t) are the available values of y[t) , pt) and u(t) respectively at time t; we now write system (12) as follows:
X(t) = FW(t)) , Vte[tQ,T] , (13) where F denotes the fixed point mapping.
We can now define the costate relaxation algorithm with asynchronous communication using the following definitions :
DEFINITION 1 . Let q and q be two non negative integers, and
s = q mod (v) + 1 ; sf = V-q mod (v) ,
q = ?l + <79
If s > I then
s (i) (q) = Sup s < s|2/M τ_ -i 1 s ' s+ l else
s(<?) = v .
If s' < V then
has just been computed by the first processor
jx(a) = Inf sf < sf \p\r r n a s JuSt been computed by the ' s/ ' s'+l second processor
else ls'(q)=\
294 J. C. Miellou and P. Spiteri
Then, the strategy s(q) is defined by : If s(q) = ((q — 1 ) mod (v) + 1 )mod (v) then s E s(q) and q = q + 1 and if sf (q)= (v— (q — 1 ) mod(v)) mod (v) then s'£s(q) and q = q + 1 ,
where ^mod(v) indicates the congruence "modulo" V .
REMARK. The value of s(q) is then the value of the time interval index [τ, , τ, ] for which a processor is in charge of the evaluation of new components on this period of time using components values known from previous iterates; when the two processors begin at the same time their computations on the same period of time s(q) is not necessarily reduced to one element. Note, also, that the scanning strategy of the time interval by one of the processors is sequential.
DEFINITION 2. Let X(°' E DF) and the sequence X^ be defined by induction as follows: VqEN, VtEJl , . . . ,α, Vfcel,...,v :
with
iY* ' ( t ) if kts x^l\t) =' τ
^ iF^Wit)) if kEs(q)
^(q-r Xq) Λ ^q-vAq) W(t) = . . . , x. 0 ( t ) , . . . , x. ΰ WeE
d9h
(14)
where R = r (q) , . . . , r (q) is a sequence of elements in A7 , which is subject to the following conditions [14] :
V j 6 1 , . . . , α , VqEN the mapping q -> q — r .(q) verifies : «7
q-v.(q) > 0 lim (q-r.(q)) = + oo .
q^oo J
5. Convergence Analysis. Note, first, that for the parallel algorithm defined in Section 3, the process interactions are not continuous, but piece-wise continuous functions on [tn,Tj ; under such conditions, the operators arising in problem (12) are defined from the set of piecewise continuous functions into the set of continuous functions [17]. To analyse convergence of the previous algorithm we may consider a sharper splitting of the operators into 3 components (3 > ot), which in the practical case corresponds to the point decomposition, and we show that the fixed point mapping related to this decomposition is contracting for an adapted scalar norm [14] in the set of piecewise continuous functions; then applying a result of El—Tarazi [7] this previous property guarantees the convergence of the asynchronous
A Parallel Asynchronous Relaxation Algorithm 295 iterations for this particular decomposition. For the splitting corresponding to the costate relaxation algorithm, we go down to point decomposition analysis.
Let ß = 2 n + rn and consider the following spaces:
h.h-c^h' W > > v i e . . .
normed by
\y\l h = sup \yt)\ , y e E l h . h n+i
(15)
Consider now the following space : V
normed by h=F,h,h (16)
\y\ = Max \y\ , z / € E 7 . ( 1 7 ) he i,.. .,v z , / z ' ^ z
We can also define the spaces :
El,h = C[Th'Th+l[)> V Z € * + I , . . . , 2 „
normed in a s i m i l a r way as in r e l a t i o n ( 1 5 ) . For φ element of the above
En 7 , we de f ine ΕΊ and | φ | as in r e l a t i o n (16) and ( 1 7 ) . F i n a l l y , for L , rl i oo
1 £ 2n + 1 , . . . , $ , we consider also
?Z = 1 normed in a similar way (see relation (17)).
Consider also ΕΪ the dual of E« , and ( , V the pairing between E,
and E * . To each χηξ.Εη we associate the so called duality mapping of E:
Gixi] = ^ieEt; (xvh)i = KC = 'M*l · where | · | denotes the uniform norm, and || · || is the norm of EZ Jl
Let
and
E = ΊΤ E Z = l υ
X = (z/(t), <p(i) , u ( t ) ) t , X € E
and consider also a point decomposition of the operators Λ and Λ ; with the notations defined above, problem (4) can be written :
A^(X) + AZX)30 , VIE l , . . . ,3· (18)
As previously, we can associate a fixed point equation to system (18)
296 J. C. Miellou and P. Spiteri in a similar way as in relation (13); we can also define a point asynchronous relaxation method by analogous recurrence as in relation (14) (but now for ie i , . . . , e ) .
For every W£D(A)9 for every fc,Z-€l,...,ß consider now the mappings defined by :
Λ«Κ> = Μωι wk-i ' wk-i>xk>wk+i > · · · ' ω β ) ·
REMARK. When k=l, the mapping k^x-^) represents the diagonal operator of h-j(X); when ki= I the mapping Λ^(χ^) represents the interaction between the Ith and the kth subproblem.
LEMMA 1. Consider the following ordinary differential equation:
(19) ^ — + λ (χ ) = zt) , V t € [ t 0 , T ] , X£R
x(tQ) = 0. Let us assume that:
λ is a continuous function and has continuous derivative of order 1 , (20)
z is a piecewise continuous function having a finite number of discontinuity points which are denoted τ ,.,.,τ , (21)
z(t) is continuous on the right at each point T., Vie2 v-l . (22)
Then, there exists g £G(x(t)) such that: 'dx(t) /ώ > o , v t e [ t n , 7 ) . dt > y - ^ > v , . L , 0 3
PROOF. Hypotheses (20) and (21) being verified, we know classically [15] that x(t) is an element of C([t0,27]), this space being normed by
\x(t)\ = Sup \x(t)\. °° t€[t0,r]
dx On account of the properties of function z, -TT- is not continuous, but piece-wise continuous and has a finite number of discontinuity points, which are
dx T0 , . . . , τ , ; furthermore we agree to take the value of -rr for t = T by 2 v -1 & dt 1 J
taking the limit on the right. Moreover, x(t) being a continuous function, the maximum of xt) is reached in, at least, one point t£ [t , T] .
We consider several cases: (1) The maximum of x(t) is reached at the point t1 € ]τ ., T. [ ; then, we know [3] that:
g (tJ ) = 6 / · sign(x(tf )) (where 6 is the Dirac function) and y
9(t'))=Jinjt±· sign(x(t'))·
A Para l le l A s y n c h r o n o u s R e l a x a t i o n A l g o r i t h m 297
But t he func t ion x r e a c h e s i t s maximum a t t = tf, and, i n a d d i t i o n x(t) has
cont inuous d e r i v a t i v e s on ]τ . , τ . , [ , then i t fo l lows τ t + 1
§ < * ' ) - o and, consequently
<§£.*(*'> - o . (2) The maximum of x(t) is reached at the point t=T. Then we have
<5P'W>-dx(T) .
sign (xT)) dt dx x and — being a continuous function on ]τ _ , T [ , we have:
because, on the contrary, it can be verified that χ(Τ) is not the maximum value.
Similarly we have
4*1*1 < 0 if x(T) =-\x\ dt ' oo
and, in either case, we have
<ä£l..<„>„. (3) The maximum is reached at one. of the discontinuity points τ..
dx If, as previously said, the value of -j- for £ = τ. is given by the value of the right derivative, we observe that this case is identical to case number 2, and the same reasoning is valid.
THEOREM 1 . Suppose that the following assumptions hold:
VZ€ 1 , . . . , 3 , VWEE, Vxl,x/l£El , Υ ^ Ε ^ Ι ^ - χ ^ )
such that
( ^ Κ ^ ^ Κ ' 5 ^ ) >ηιι\χΓχΙιζ» ηιι>0)> \/l,k£ l, . . . , 3 , k ^ l , \/W£E, V x f e , x ^ ^
(23)
(24)
Ik^k1 "ik^k1]^^ "Ik^k ^Vo* K ylk
V i e 2 η + 1 , . . . , 3 , V χ χ' ΕΕΊ , ν η € Af (x
ν η ^ Λ ^ ) , 3 ^ E G z ( x z - a ; [ ; (25)
such t h a t
/η-7 - i f - , , ί7-,' Ί ( " ΐ - « 1 ' 9 ΐ \ > 0 ' and, if furthermore the matrix N~ ( 77,) is an M-matrix, then any asynchronous point iteration, applied to the resolution of system (18), and starting
298 with a vector X (0)
J. C. Miellou and P. Spited converges to the unique solution of the problem.
REMARK. Assumption (23) corresponds to a property of strong accretivity of operator Λ77 ; assumption (24) explains a Lipschitz condition of interaction mapping; assumption (25) corresponds to a property of w-accretivity of operator 3ψτ, ; note that this hypothesis is in our case verified because the
ad above operator is not decreasing [17].
PROOF. Let x* denote the exact solution and X-, be the value of the I
component obtained by the algorithm. Let, also, W denote the values of the interaction variables between the I -equation and the others; W is a piece-wise continuous function having a finite number of discontinuity points.
We can write the equation verified by #- and x* respectively; then, we subtract these relations and we multiply the difference by #7 € G7 (.
We consider three cases: 1st case For Z-Gl,...,n, we have
d 1 _ * < dt Xl xl'
xl xl]
Η/τ + (V*>-¥**>-*z>i = °'
where w,
But, we may write
h^~Al p. f
w l - i > x l
V*t Z + i ' '
x k - i ' w k ' w k + i , wQ)
k=i
and we have
where
Consequently w = \χ\
Λ / * * V x i " · · ' xk ' wfe+i '
,k
• v , x k-i
•k
k £
' xk-\ ' xk ' wk+i
A?*<**> 3 lkx k'
, # fc-i , ZJ fe+i ·
and
A7(X)-KZ(X") , gr
d t l Ä 7 ' V ' ^ Z
fc=i
Λ ^ (*7 11 L
'ϊκ^
*ϊιΙ*ϊ
Alkxk)
·**>,
fc = i
Λ«Κ> Λ W I * \ Aik χι]> Η
(26)
On account of assumption (23) and by Lemma 1 the left hand side of relation
A Parallel Asynchronous Relaxation Algorithm 299
(26) is greater than ηιι\χι~χη\ 5 furthermore, by Cauchy-Schwartz inequali
ty, we have
k*i η k*i k=i k = \
and owing to
Then Ik; IL = l^-**1
fc = l
w h e r e rc.., j =
l k
Ik nu '
2n d case: For I € n 4- 1 , . . . , I n , by an appropriate transformation of the variable t, of the type Θ = T - t , we observe that this case is identical to the previous case, and the same reasoning holds.
3rd case: For 1 E 2n + 1 ,...,3, with respect to relation (25), and by repeating the reasoning already developed in the study of the first case, we obtain the same type of inequality. We can then write
3 l*z"**'«,* Σ, Jik'^wk"xkL9 v z e i , . . . , e , (27)
k = i k^l
where J is the Jacobi matrix associated with the matrix N; it is a non-negative matrix, with spectral radius pj) < 1 . By applying the Perron-
v) R -Frobenius theorem, we know that there exists a vector Γ € R , with strictly positive components, such that
JTV < vT V , v€ [p(j) , 1 [ ; (28) then R I _ *j
\x -or* I <( V J · T V V Max ( R * Η " Λ lxi xiL^ v L ik [kj M*x v r v j -
Zc=i K k
Considering relation (28) and on account of the definition of | · | given by relation (17), we may write
\XT-X*\ < νΓ^ · Max ' h
Max \wk-x\\Kh
L k * l \ ΤΪ
300 J. C. Miellou and P. Spiteri
The inequality may be written :
/M*x \χι~χι\i,h\ (™a*\wk-xl\k.x Max ) < V Max
* e i , . . . , 3 \ ^ J fc€i,...,3\ r
which defines a property of contraction on the space E; after El—Tarazi [7], this contraction property guarantees the convergence of asynchronous point iterations.
COROLLARY 1. If assumptions (23), (24) and (25) hold, then the costate relaxation algorithm with asynchronous communication, described by relation (14), converges to the unique solution of problem (4), for every starting value
7^. PROOF. We consider the decomposition (9) of problem (11) in a(=3) blocks; we perform as previously by writing the equations verified by one iterate denoted x. and those verified by the exact solution x?, according to the notations defined in Section 3; from the block splitting, we go down to point decomposition and then in the proof of Theorem 1, we replace the contracting matrix J by a contracting matrix J associated with the block decomposition of the matrix N defined in relations (23) and (24); then we obtain a block contracting matrix and the same reasoning carried out in the proof of Theorem 1 can again be made here [17]. 6. Application to Optimal Control Law Computation of a Thermic Process. The example is connected with an optimal control law calculation for a vertical oven with three heating zones [9] which has six state variables and three control variables; a study with linearization around a chosen point gives the linear model :
-rr + Ay = Bu , ytQ) = 0 and z = Cy ,
where z is the output vector and A is an M-matrix. The performance index to be minimized is :
T
where the parameter k controls the comparative weight given to the two components of the cost function: accuracy and energy expenditure. The term Un
is the steady-input which should be applied to obtain the desired output z n
after an infinite duration. Furthermore, the control vector components must verify the following
constraints :
A Parallel Asynchronous Relaxation Algorithm 301
w. < w . < w . , vte i,...,3 , vte[t , r ] . Application of the theoretical result obtained in Theorem 1 and Corol
lary 1 shows that if k is greater than a critical value, denoted k , then the costate relaxation method and the parallel costate relaxation method with asynchronous communication converge [17]. Numerical experiments showing that the use of the costate relaxation algorithm can be very efficient compared to other classical algorithms have been carried out in a sequential context [17]. On the other hand, the parallel costate relaxation method has been tested using the simulation of parallel executions; when two processors are used the speed-up obtained equals to 1.4 or about; furthermore the conflicts of access between processors are few and correspond to 0.04% of the total time of computation. Finally, the duration of communication between the processors is important; however, algorithm efficiency is not perturbed and can even be improved since the above communications allow more efficient computation [17].
7. Extension to more General Cases. One possible extension of the previous study corresponds to the case of problems in which the objective contains a function of the terminal state :
T
J = K(Υ,Τ) + J ry,u) dt. *0
The Hamilton-Pontryagin minimum principle leads to the necessary optimal conditions with the same shape as equation (4) excepting the value of the costate at time t = T given by
P(T) - If . This problem can be reduced to the form of system (4) by appropriate transformation of the following type :
p(t) =?(*)-|f. vte[t 0,r]. Then, if assumptions (23), (24) and (25) hold the studied algorithms applied to the previously problem converge.
Note that the above theoretical study is again applicable in the case of singular perturbations in control theory and also of control problems with several objective functions (see [8]).
REFERENCES 1. V. Barbu, Non Linear Semi-Groups and Differential Equations in Banach
Spaces , Noordhoff International Publishing, 1976. 2. G.M. Baudet, Asynchronous iterative methods for multi-processors,
Journal of A.CM. 25 (1978), 226-244. 3. Ph. Benilan, Equations Drevolution dans un espace de Banach quelconque
et applications, Thesis, Orsay, 1972.
302 J. C. Miel lou a n d P. Spi ter i
4. D. Chazan and M. Miranker, Chaotic relaxation, Linear Algebra and its appl. 2 (1969), 199-222.
5. P. Comte, Iterations chaotiques ä retards. Etude de la convergence dans le cas d'un espace produit d*espaces vectoriellement normes, CRAS, serie A, 281 (1975), 863-866.
6. P. Comte, J.C. Miellou and P. Spiteri, La notion de H-accretivite-Applications, CRAS.. serie A, 283 (1976), 655-658.
7. M.N. El-Tarazi, Some convergence results for asynchronous algorithms, Numer. Math. 39 (1982), 325-340.
8. D. Gien, B. Lang, J.C. Miellou, L. Raffort and P. Spiteri, Commande optimale de systemes complexes, RAIR0-. 18 (1984), 209-224.
9. B, Lang, A.N. El Awtani, P. Spiteri and N.E. Cheikobeid, Decentralized calculations in optimal control of a large thermic process: methods and results, 2nc* IF AC Symposium "Large Scale Systems"', Toulouse, 1980, Pergamon Press, 505-516.
10. B. Lang and P. Spiteri, Decomposition and coordination using asynchronous iterations in Optimal control, Encyclopedia of Systems and Control (Ed. M. Singh), Pergamon Press, 1983.
11. J.L. Lions, Controle optimal des systemes gouvernes par des equations aux derivees partielles, Dunod, 1968.
12. J.C. Miellou, Methode de l'etat adjoint par relaxation, RAIRO - R\ > (1972), 81-87.
13. J.C. Miellou, Problemes de controle optimal H-decentralisable: la methode de relaxation —decentralisation, CRAS. serie A, 277 (1973), 609-612.
14. J.C. Miellou, Algorithmes de relaxation chaotiques a retards, RAIRO — R\ , (1975), 55-82.
15. L. Pontryagin, V. Boltianski, Gamkrelidzer and E. Michtchenko, Theorie mathematique des processus optimaux, Edition de Moscou, 1974.
16. F. Robert, Contraction en norme vectorielle : convergence d'iteration chaotique, Linear algebra and its applications 13 (1976), 19-35.
17. P. Spiteri, Contribution a, I1 etude des grands systemes non lineaires, Thesis, Besan^on, 1984.
18. A. Titli, Commande hierarchisee et optimisation des processus complexes, Dunod, 1975.
ON THE VALIDITY AND STABILITY OF THE METHOD OF LINES FOR THE SOLUTION OF
PARTIAL DIFFERENTIAL EQUATIONS
M. N. MIKHAIL Department of Computer Science,
The American University in Cairo, Cairo, Egypt
Abstract. The method of Lines (MOL) for the numerical solution of partial differential equations is presented. MOL application to equations of the elliptic type is examined in view of clarifying that the instability usually blamed on MOL has its origin in the original equation. MOL system of ordinary differential equations is shown to be a valid representation of the partial differential equation. The effect of increasing the number of lines used is discussed. It is shown that as the number of lines is increased the accuracy of the MOL representation to the original system increases.
1. Introduction. Partial differential equations frequently appear in physical sciences and engineering applications. The analytic solution of these equations is possible in simple domains with specific boundary conditions which limit its use only to problems of academic interest. For more general conditions that are frequently encountered in practical situations, we usually have to seek solutions in numerical form.
Among the possible numerical methods, the Method of Lines (MOL) is particularly interesting. The advantages of MOL are discussed by the present author in [l]. Two specific advantages are worth mentioning here. The first advantage is that MOL transforms the partial differential equation into a set of coupled ordinary differential equations. The solution of ordinary differential equations is well established and can be classified as routine work at the present time for which a number of accurate, stable and well-tested algorithms are available. The second major advantage of MOL is that the number of unknowns is an order of magnitude less than in the popular finite difference methods for the same accuracy.
303
304 M. N. Mikhail One main disadvantage, which has been consistently blamed on MOL, is
that it is unstable when applied to equations of the elliptic type (see [2] for example). One of the main objectives of the present work is to point out a fact that has been missing, namely, that the instability is inherent in the nature of the elliptic equations themselves and should not really be blamed on MOL.
In spite of this instability, the Method of Lines has been successfully applied to a number of elliptic problems of practical importance. For example, MOL has been used to solve the flow equations in ducts with arbitrary shape [3], to solve the supersonic blunt-body problem and conical flow problems [4], to solve eigenvalue problems [5], to solve the elastic rectangular plate problem [6] and is used to solve the Poisson's equation with nonlinear and free boundary conditions [7].
The Method of Lines is stable and simpler in application when it is used to solve hyperbolic and parabolic equations. It has been used extensively to solve these equations since its first appearance in 1930 [8],
The Russian work in connection with MOL was reviewed in 1965 [9], A general review of MOL was carried out in 1968 [10], in 1972 [2], in 1978 [11] and in 1982 [12].
The purpose of the present work is to establish a relation between the MOL solution and the analytic solution of a classical elliptic equation, namely, Laplace's equation in two dimensions and point out to the origin of the instability in the solution and the effect of increasing the number of lines used.
2. MOL Applied to Elliptic Equations. For the purpose of the present discussion, let us consider Laplace's equation in two dimensions:
^4 + —"- =0, (1) in the square domain: 0 < x < 1 , 0 < 2/ < 1, subject to the following boundary conditions:
u(x9Q) = ux, 1 ) = 0 (2)
u(09y)=fQ(y) (3)
u (l , y) = f 1 (y) · In order to solve this problem using the method of lines the domain has to be divided into strips by equally-spaced /l/-lines parallel to one of the axes, say the χ-axis, as shown in Fig. 1.
Validity and Stability of the Method of Lines 305 2 / | 1
n
N
2
J = 1 u J=0
FIG. 1 The computation domain
The unknowns of the problem are now:
where h-N+ 1
U (x) = u(x9jh) , j = 1,2, . . .,N
The differential equation (1) can be written as:
u = — u xx yy (4)
and the right-hand side can be replaced by an appropriate finite difference approximation of the form:
d2U -\ U . , - U . + U . . . + E , (5)
where E is the truncation error. In this case,
with 24
u . Ί <u* < u ... .
Using (5) in (4), we get the set of N second-order ordinary differen tial equations:
d2U. =ΤΪ^.7-1-2^ + ^ + 1 ] 5 J' = l.2. ···.*·
Note that, in this case, UQ = 0 and UN + l = ° represent two of the boundary conditions.
Let dU ./dx =V. . Equation (6) can be written as a set of 2/1/ first-J d
order ordinary differential equations of the form:
(6)
306 M. N. Mikhail
_d_ dx
i . e . ,
dx W = AW,
where W i s the v e c t o r of unknowns
W =
2/1/X 1 and A is the coefficient matrix given as :
2NX2N
1 /z2
2
- 1
0
0
- 1
2
- 1 2
0 0
0
Al
0
-1
I 0
2/1/ X 2/1/
0 0
0 0
0
0
0
0
or A. = Γ a · 1 > w n l L ^ , 7 J
a . . 1-0
+ 2//z for i = j
-\/h2 for K - j | = I
0 otherwise .
(7)
(8 )
(9)
(10)
Validity and Stability of the Method of Lines 307 The MOL solution is obtained by integrating the system of equations
(7) subject to the boundary conditions:
^(0) = f0ih) , i = 1,2,3, ...9N (11) and
^ ( 0 = f^ih) , i = 1,2,3, ...,N. (12)
The solution is normally carried out numerically using one of the shooting techniques, i.e. start at x = 0, ί/.'s are defined by equation (11) and the unknowns 7.fs are chosen to satisfy the condition at the other boundary (equation (12)), namely, that i/.(l) = f^ih) .
The stability of the numerical procedure with respect to small errors that might be introduced as a result of numerical roundoff for example can be judged by the eigenvalues of the matrix A. Negative eigenvalues indicate stability while positive eigenvalues signal instability.
3. Relation between Analytic Solution and MOL Solution . (a) The Analytic Solution. The analytic solution for
32u . d2u + -£= 0 is known to be:
u = ) u .
i=i with
u . = \ a .
ei (equation (2)), i.e
e +D.e ^
(13)
sin kry , (14)
where L's are the eigenvalues required to satisfy the boundary conditions
sin k . = 0 then
k. = iv , i = 1 ,2,3, . . . ,oo . (15) The constants a.'s and Z?.fs are chosen to satisfy the boundary conditions:
u0 ,y) =fQ(y) and u( 1 , y) =fY y) .
There are two important remarks concerning the above analytic solution. First. Although the analytic solution (13) is an infinite series, we have to consider only a limited number of terms in practically computing the value of u at any point. Second. Exponentially growing terms exist in the analytic solution. This means that even the analytic solution is unstable with respect to errors in defining the coefficients a.'s. Such errors get magnified by the factors
k- x e i .
308 M. N. Mikhail (b) MOL Solution. The MOL solution is the solution of the system of
equations (7). The square matrix A is of order 2/1/; it possesses 2/1/ eigenvalues, i.e., the MOL system would approximate the solution of the original equation by means of a linear combination of a finite number (exactly IN) of linearly independent solutions. The corresponding eigenvalues are those of the matrix A (equations (9) and (10)).
The characteristic equation of A is given by :
det [Α-λΐ] = 0 i.e.
-XI I 1 det L A, -XI)
= 0
Since A, and I commute, the above equation can be written as:
det [A -X2l] = 0 . 2 So, if we define λ = λ ,
det [Al ~Χλΐ] = 0 , i.e., X's are the eigenvalues of A .
This means that the 2N eigenvalues λ of A are defined as:
λ ^ ι · The N eigenvalues of the tridiagonal matrix A (equation (10)) have
been calculated [12,13] to be: /Λ \ 4 . 2 ( iv \ k I'* h2 V2(/l/+l)/
Hence, the 2/1/ eigenvalues of A are: 2
1 , 2 , . . . , / ! / .
λ . \2( / l /+ \)J ' 1 , 2 ,
Then, the MOL s o l u t i o n would b e : N
vi = Σ i=\
X η· x C . . e + D . . e
-X .xi ^
,/!/ .
!,...,/!/,
(16)
(17)
where U.= U(x,jh) and the coefficients C . . and D. . are such that the boundary conditions at x = 0 and x=\ are satisfied.
(c) Relation between analytic and MOL solution. In comparing the analytic solution equation (14) with the MOL solution equation (17), we find that the MOL solution approximates the analytic solution by using only the first N terms where
and C . . = A . sin k . y> ^y^ ^ τ 3
D . . = B . sink . y . 2-, J i is 0
1,2,...,^/
where X.'s of the MOL solution correspond to k .'s of the analytic solution.
Validity and Stability of the Method of Lines 309 For MOL solution to be valid the A.'s should converge to the exact
k/s as N increases. This is what we should examine next. ^
4. Effect of Increasing the Number of Lines N . The effect on the Eigenvalues A.'s: For specific number of lines N
using equation (16) which can be rewritten as:
λ . = 2(N+ 1) sin iir ] 2(N+ 1) J ' 1,2, . . .,/!/.
The effect of increasing the number of lines N on the first ten A.'s to the analytic values k . as defined by equation (15) are shown in Table (1).
Table 1. MOL Eigenvalues A. N.
t
3 4 5 10 50 100
K.
1
3.06 3.09 3.11 3.13 3.14 3.14
3.14
2
5.66 5.88 6.00 6.20 6.28 6.28
6.28
3
7.39 8.09 8.49 9.14 9.42 9.42
9.42
4
-9.51 10.39 11 .89 12.53 12.56
12.57
5
--
11.59 14.41 15.65 15.69
15.71
6
---
16.63 18.74 18.82
18.85
7
---
18.51 21 .82 21 .95
21 .99
8
---
20.01 24.88 25.07
25.13
9
---
21.11 27.91 28.18
28.27
10
---
21.78 30.92 31 .29
31.42
Note that the expression for A . can be expanded as follows :
A . = 2(/17+ 1 s . Γ ii\ 1 ) sin
L2(/17+1)J = ?(N+ i J ^ if I* V + _L_ f_il_
i V ' L 2 ( / l / + D 6 V 2 ( / 1 / + 1 ) / 120 ^2(717 + 0
24(/l/+ l ) 2 120X 24(/l/+ l)*4
As /!/->«>, A . = iv , i.e., the eigenvalues A. tends to their exact values for large number or lines N.
The effect of increasing the number of lines N on MOL solution can be summarized as follows: (1) The number of eigensolutions that MOL contains is equal to double the number of lines. Hence, increasing N makes the MOL solution more representative of the original problem. (2) Increasing the number of lines improves the accuracy of the eigenvalues. (3) As the number of lines increases, the number of boundary points (the
310 M. N. Mikhail ends of each line) considered increases, resulting in a better representation of the boundary values. (4) The eigensolutions included as the number of lines is increased are associated with higher eigenvalues. This inclusion, however, makes the solution more sensitive to small errors, i.e., more unstable. As indicated earlier, it is important to note that these high eigenvalues belong to the original problem and are not introduced by the method of solution.
5. Conclusion. From the above discussions, we can conclude that the MOL system of equations is a valid representation of the elliptic differential equations (Laplace's equation in the present case). The instability built in MOL has its origin in the original equation.
REFERENCES
1. M.N. Mikhail, A super-element approach for the solution of problems of the boundary value type, Proceedings of First International Conference on Theoretical and Applied Mechanics, Cairo, Egypt (1980), 12-22.
2. D.J. Jones, J.C. South, and E.B. Klunker, On the numerical solution of elliptic partial differential equations by the method of lines, J. Comvut, Phys. 9 (1972), 496-527.
3. M.N. Mikhail, Optimum design of wind tunnel contractions, AIA A J. 17 (1979), 461-477.
4. E.B. Klunker, J.C. South, and R.M. Davis, Calculations of nonlinear conincal flows by the method of lines, NASA TR i?-374, (1970.
5. D.J. Jones, and J.C. South, Application of the method of lines to the solution of elliptic partial differential equations, Canadian NEC No. 18021, 1971.
6. R.C. Thompson, Convergence and error analysis for the method of lines for certain nonlinear elliptic and elliptic-parablolic equations. SIAM J. Numer. Anal. 13 (1976), 27-43.
7. G.H. Meyer, The method of lines for Poissonfs equation with nonlinear or free boundary conditions, Num. Math. 29 (1978), 329-344.
8. E. Rothe, Zweidimensionale parabolische randwertaufgaben als grenzfall eindimensionalle ranswertaufgaben, Math. Ann. 102 (1930), 650-670.
9. O.A. Liskovets, The method of lines (review), Differential Equations, 1 (1965), 1308-1323.
10. N. Tam, Application of the straight lines to irregularly shaped regions, USSR Comp. Math, and Math. Phys. 2 (1968), 124-138.
11. Z. Aktas, On the application of the method of lines, Dept. of Comp. Sc. Report, Middle East Technical University, Ankara, Turkey, 19*78.
12. A. Sharaf, Stability of the method of lines for numerical solution of partial differential equations, M.Sc. Thesis, Dept. of Engineering Math., Faculty of Engineering, Cairo University, 1982.
13. M.N. Mikhail, and A. Sharaf, Stability of the method of lines for elliptic equations, 7th International Conference for Statistics, Computer Science, and Demographic Research, Ain Shams University, Cairo, Egypt (1982), 177-186.
CAUSALITY IN ALGORITHMS AND ARCHITECTURESt
W. L. MIRANKER and A. M. WINKLER IBM T. J. Watson Research Center, Yorktown Heights, New York, USA
Abstraet. Constituents for a theory of computer architecture design intended for parallel applications are presented. The constituents are causality, dependence and scheduling in algorithms and in architectures. Algorithms are classified and then described by means of a class of diagrams. Architectures are described by a space-time model whereby causality and dataflow in the physical computer is characterized. A principle for scheduling is introduced and a number of case studies are made. The studies reveal several developments for a theory of computer design which follow from our approach. Moreover it is seen that scheduling methodologies in graph theory could be applied to parallel computer design.
1. Introduction. The development of new computer architectures often proceeds with the objective of executing particular algorithms more effectively. Formalization of this process would constitute a theory of computer architecture design. In this paper we frame an initial form of such a theory.
We describe two classes of relevant objects: (i) algorithms and (ii) architectures in abstract terms, and discuss mappings between the two classes. Our method is motivated by the approach in our previous work. There [4,5] we describe a method for computing certain kinds of programs by using a type of computer processor hardware called systolic arrays. Basic to this was the construction of certain mappings. These mappings correspond to schedul
ing the execution of each step in the program on a particular processor at some moment of time.
Invited talk.
311
312 W . L. Miranker and A. M. Winkler In this way the space-time representation of the process of computation
comes to the fore. We noted that the key concept was graph-theoretic, and here we explore the consequences of this viewpoint: that the same framework that characterized the systolic case should admit generalization to arbitrary algorithms and architectures. This introduces the methodology of scheduling in graph theory as a basis for solving problems in parallel computer architecture.
One might hope to create a procedure which would accept as input a machine architecture together with some formal specification of an algorithm (e.g. a program), and which would output a dependence graph for the algorithm, with an optimal schedule of the algorithm in the given architecture. In fact, we shall see that this goal cannot be attained. Nevertheless, structure emerges which sheds light on the design of parallel architectures.
Moreover, as essentially different types of algorithms are studied, new architectures should emerge from the corresponding dependence structures and schedules.
We begin in Section 2 by introducing the language of partially ordered sets to highlight the concept of causality. A notion of scheduling a computation can be framed through this concept.
In Section 3 we describe the class of algorithms which can be executed on a computer through use of recursive function theory. We show how a diagram may be generated corresponding to any relevant algorithm. The diagrams are exploited for discussing dependence in an algorithm.
In Section 4 we introduce a space-time model of a computer architecture and formalize causality and data flow in the physical computer.
In Section 5 we employ the framework introduced to define and characterize the scheduling of algorithms in architectures. Examples of techniques for defining schedules are described.
In Section 6 we present three case studies of application of our methodology: (i) linear algebra (systolic arrays) (ii) cyclic reduction and (iii) binary search.
2. Schedules. From our study of the systolic case, the importance of focusing on systolic isomorphism as a principle relating algorithms to architectures is clear. Moreover, a way to generalize this mapping principle is clear; these maps are simply 1—1 order preserving maps, i.e. embeddings of (partially) ordered sets. We recall how these structures arise in the systolic case, departing slightly from the view point of our previous work.
We consider a program P (consisting, for convenience of a nest of
Causality in Algorithms and Architectures 313 FORTRAN DO-Loops) and examine the dependence among the program statements A,B ... . We say that one statement depends on another if the latter generates a value which is referenced by the former. When we perform this program on a systolic array of hardware,A must be performed early enough so that its output has time to reach the processor where B is scheduled to be performed before the time scheduled for the execution of 5. We can formalize the order defined by dependence in the program loops, and we can also formalize the order defined by time and causality in the physical process of computation. The first characterizes the relations between steps in the algorithms and the second mirrors the physical flow of information in the chosen architecture.
By a (consistent) scheduling we mean an assignment of a processor and a time to each program step which respects the causal structures (i.e. dependences and time orders) thus defined. The condition for consistency is simply that order relations arising on the algorithm be preserved in the hardware execution in space-time.
In our prior work we note that an algorithm has a graph structure, which is directed and acyclic (since no step of an algorithm could depend on its own output). In implementing the mapping principle, it is useful to take the transitive closure of this directed acyclic graph, and to suppose that the graph is reflexive. It is known that a (partially) ordered set is precisely the same structure as a directed acyclic transitive reflexive graph. See [6] for details.
In order to highlight the concept of causality, we continue our development employing the language of partially ordered sets. Causality defines a natural order both in the structure of programs and in the physical structure of space-time.
3. Algorithms. There are several (computably equivalent) ways to describe the class of algorithms which can be executed on a computer. These include recursive function theory, formal grammars, programming languages, machine languages and Turing machines [3].
We seek to attach to an algorithm a partial order, which characterizes the dependence in the algorithm. We can define the dependence in the following way. Consider a program P and all possible programs obtained by permuting the steps of P. (Of course not all permutations of P are valid programs). Let P be the set of all such programs that are equivalent to P, in the sense that they compute the same function as P. We can partially order the statements Α,Β,... of P by intersecting the orders of the elements of P. That is, B>A if S precedes A in every program in P.
314 W. L. Miranker and A. M. Winkler
This nor any other definition of dependence is computable [l]. The framework of recursive function theory offers insight into the ob
structions to constructing dependence relations in algorithms. In spite of these obstructions, in many cases of interest, we can construct dependence relations and explore the resulting schedules in space-time.
We recall that the class of primitive recursive functions is the collection F of functions f:N -* /IT generated by the following rules.
1.
2.
3.
4 .
5.
6 .
8:N->N; x + > a ; + 1 6 f ' t h e successor f u n c t i o n '
z:N^N; x ^ O E F ' t h e zero f u n c t i o n ' k π . : N ->N; (x, , . . . , χΊ ) +* x . € F ' p r o j e c t i o n s '
/ : iT->tf2? € F , g:N ->Nie1:^fXg: 1T^NP+ £ F ' c l o s e d under p r o d u c t s '
f :Nk - > / e F , g :tiF -+N1 € F=> g°f:Nk^>N € F ' c l o s e d under compos i t ion '
g:Nk->N£F, f :Nk + 2 ^N € F =>h : Nk+ l -» N ;
hx ,n) = [g(x) i f n = 0if(n9x9h(x9n— 1)) i f n > 0] G F
'closed under induction' ,7^+1 /1/€F the following To these rules we add, for each function g: N
function: 7. f = (x , n) | £7θ, n) = 0, and g(x 9m) =0=>m>n
to obtain the class of (partial) recursive functions. We construct a certain (data flow) diagram for any function in F .
These diagrams will be useful in discussing dependence. For h:N -*N this diagram has the following form
K COPIES
N*—-iÜ-^N
^ Ν * ^ N
COPIES
The arrows on the left are inclusions maps, meaning that the ^ ,-th ,_ ^c u flK
•th copy of N
maps identically onto the i component of each We regard these left-hand copies of N as inputs, and the righthand ones as outputs. We will sometimes abbreviate this as:
k COPIES r COPIES
Causality in Algorithms and Architectures
For example if h = s , given in rule 1, we have the following diagram
315
N-
For h = z, given in rule 2, we have the following diagram,
N N
which shows the output to be independent of the input. If h = π 7 . : N -* N; (χΛ, . . . . x ) +> x . as in rule 3, we have
rM -N
which becomes
i.e. the output depends only on the it input. Note that the total number of outputs is r + I. Similarly let the dia
grams for f and g be
k COPIES N-
N H
|—N r COPIES
KN I | NH • )jt COPIES
Then corresponding to rule 5, we get a diagram for gof by identifying the outputs of / with the inputs of g:
N H h-N-
K— N-
h-N
l-N
For the diagrams corresponding to rules, 4, 6 and 7, see [6],
316 W. L. Miranker and A. M. Winkler We can reduce (simplify) diagrams somewhat by a pruning process. In
addition to the original output of the total function, there will be extraneous Outputs' or /l/'s with no arrow leading from them. We eliminate these along with all the arrows leading to them. This may introduce new extraneous outputs, which we again eliminate, together with the arrows leading to them. Continuing, we will eventually have a diagram with no extraneous outputs.
Example. Consider the projection i\l : N ->N, which has the diagram:
2 The node N is extraneous, so we eliminate it as well as arrows leading to
N
We are left with an extraneous copy of N which we eliminate, giving as a final diagram:
N -i> N
We view these diagrams as representing the flow of information from inputs to outputs, and a given step depends only on those 'preceding' steps which eventually lead into one of its inputs. We can formalize a causal structure on the arrows of the diagram (where we emphasize that an arrow is a primitive computation, e.g. a step in a program) by saying that an arrow depends on all those arrows that lead into it, i.e. A >B if there is a chain of arrows A=A,,A2 , . . . , A,, so that the output of the arrow A. is an input of A. and the output of A-, is an input of B. That is
Α=Αχ Α 2 A , B > · > i—> · > =» A>B .
Clearly B can only depend on A in the sense we discussed earlier if A> B,
but unfortunately, the converse is not true. The reason for this is that there are many ways of expressing the zero function 2 as a recursive function, and there is no recursive way of testing if a function is zero. Thus, for example, we could have a diagram
Causality in Algorithms and Architectures 317
N — N ! N2
Ik yf N ! N 2 '
where it seems that g depends on /. But if k= 0, then g really doesn't depend on /, so f and g could be computed in parallel. Dependence is very sensitive to the form in which an algorithm is expressed, and the causal structure can be expected to provide sufficient conditions for consistency (i.e. for the specification of a consistent schedule): usually not necessary ones.
4. Architectures. We view a computer as a collection of processing units (which might be processors, memories, caches, registers,. . .) connected by wires, and we describe the data flow in such a construct. The key concept of time allows us to formulate a description of data flow by using the idea of causality.
In physical space-time we construct an ordering which specifies 'domains of influence', and which corresponds to the possibility of passing messages. The points to be ordered must correspond to the performance of a calculation, in turn corresponding to a choice of a processor 'where' the calculation will be done, and a time 'when' the calculation will be executed. Thus the points to be ordered are the points on the worldlines of the processors and memories.
We say (Α,τ) > (B ,τ') (i.e. (Α,τ) can influence (B , τ ' )), where A,B
are processing units, and T,resp. τ/ is the time parameter on the worldline of A, resp. B , if a message sent from A at time τ along the wires connecting A with B can arrive at B by time T; , This amounts to saying that the event (A9T) can influence the event (β,τ')·
We can make our space-time model of a computer architecture more realistic in several ways, incorporating some of the following ideas.
1. Some calculations take longer than others, and some processors are faster than others. We can account for this by associating a time interval T. (p) with every process or primitive computation p and each processor A. T, (p) is the time required by processor A to execute p. Then we can model the time required for processing by allowing no messages (i.e. no computed results) to leave A until a waiting time of length at least T (p) has elapsed after the arrival of the inputs required for p. This concept is limited by the fact
318 W. L. Miranker and A. M. Winkler that the time to perform p on processor A may depend on the magnitude of the inputs that p acts on; magnitudes which cannot be bounded a priori. Of course, every computer has a largest representable number beyond which overflow occurs, but setting T.(p) equal to the largest possible execution time required in every case would in effect enforce synchrony. Synchrony would normally hinder parallel performance.
2. The channel capacities between processors are finite, so information theory should indicate additional delays or restrictions.
3. We can add the concept of memory. Not all the causalities described above are realistic, because a message sent from A at time τ might arrive at B at time τ' , and by definition, (Α,τ) > (Β,τ'). Moreover (Α9τ) > (5,τ; + τ) for all τ > 0. But this is unrealistic, since a processor can't store results waiting for a later time, at least not for a large number of inputs. This is the function of memory. Hence we can change our causal relation to say (4,τ) > (Β,τ') when a message sent from A at time τ will arrive at processor B at precisely time τ', and in addition, for each memory A , (A , τ) > (A , τ + t) , t > 0. This assumes an unlimited capacity for the memory A,
4. We could add a short-term cache to processors, so that (Α,τ) > (Α,τ+t)
for 0 < t < T. However this causes loss of transitivity for the order. (Whenever we take into account the finite capacity of a system we are faced with 'overflow* problems.)
5. Scheduling Algorithms in Architectures. At this point our general model consists of an ordered set, or causal relation Γ corresponding to an algorithm and also a causal relation Ω corresponding to a (space-time) architecture. Note again that the causal relation corresponding to the algorithm may turn out to be more restrictive than necessary. (Recall the example in Section 3).
A (consistent) schedule is a map η: Γ-*Ω which is 1-1 and which pre
serves the causal structure. That is A > B =>r)(A) > η(Β) where we recall that r\(A) € Ω is a processor at a specific time.
If we suppose Γ,Ω as given, it is natural to ask questions about possible schedules η, such as do they exist, is there an optimal one, can it be constructed.
Causality in Algorithms and Architectures 319 Note that the minimal possible time to perform an algorithm correspond
ing to Γ is bounded below. The bound is dependent upon the length of the longest chain A, > A^ > . . . > A, in Γ. (The length of a chain is the number of distinct elements in it.) For if we have such a chain and if T(A.)
is the time required to perform A . then r\(A^) > r\(A2) > T)(A^) ... is a chain in Ω which spans a time at least as great as Τ(Α^) + .. .+ T(Aj.). But then it certainly takes time at least T(A.) +. . .+ T (An.) to perform Γ, which is a lower bound of the anticipated kind.
On the other extreme, we could choose a total order consistent with the causal structure Γ, and perform Γ serially on a single processor. Thus there is no communication time, and the time required will be
V L T(A) ,
AET which is hence an upper bound for the minimum scheduling time.
Suppose that communication is instantaneous, that a large collection of processors is available, and that T(p) = 1 for any p, then the lower bound is achieved, as follows. Let I\ = p € Γ | ρ = ρλ > p 2> . . .> p- . Pi + i for some i < k Γ, is the set of elements p such that the longest chain leaving p has length less than or equal to k. If T is the length of the longest chain, then Τφ=Τ9 while Γ. is precisely the collection of outputs, i.e. no further calculation depends on them. This partitions the set Γ into sets T,=T,—T,_ of those elements for which a maximal chain has length precisely k .
Example. A triangular solver gives rise to a relation Γ of the following form.
320 W. L. Miranker and A. M. Winkler We label this diagram by numbers, 1 , 2 , . . . ,9 corresponding to T, , T„,... Tg .
We can separate this diagram by 'contours' as follows.
If we let k (or rather 9—k) be the time, we can reduce this figure in a more useful form.
♦•o
t-l
t»2
t-3
♦ •4
t«5
t-6
t-7
«•a
Causality in Algorithms and Architectures 321 It is now clear that we can schedule this algorithm consistently by
putting any processor to work at the initial moment on node 9 at the level t = 0, then at the next moment letting any processor compute node 8 at the level t=\9 then picking any 2 processors to do the two computations at the level t = 2, (for nodes 7a and 7b) and so on.
This schedule maps 2\ to the space-time hypersurface t = T—k. Thus we schedule η: Γ-+Ω by letting η(ρ) be any processor at all, at time T — k
if p € I\ but p £ Γ\ .. . It is easy to see that this schedule is consistent. (See [5] for additional details.)
We call the idea underlying this scheduling the P-principle. According to this principle, first fix a deadline (T) and then schedule everything as late as possible consistent with meeting the deadline. This heuristic seems to be useful in constructing η in practical cases, and it has been used independently since the dawn of time, procrastination.
However it is clear that a good scheduling scheme, recognizing finite resources and communication costs, would pace work so as to do tasks which can be performed earlier or later when processors would otherwise be idle.
Actually we can get slightly more from this example. We can ask what connection (wires) there must be in an array which meets the requirements of this schedule. As a space-time diagram we have
where the boxes denote processors and the vertical (dashed) lines represent their worldlines. The wires emerge if we just 'project' the space-time diagram onto space, by collapsing worldlines to points. MAA—v
322 W. L. Miranker and A. M. Winkler
D D D D D Here, a simple hardware configuration has emerged in a natural way from our considerations. Deeper examples of this will be given in Section 6.
6. Fundamental Algorithms and Architectures. It seems clear from the close relationship between the causal structures of algorithms and of architectures, that algorithms should give rise to architectures. Indeed many interesting architectures which have been proposed are related to important algorithms.
We consider some case studies: Case Study 1: The computations of linear algebra often have a regular structure such as a nest of loops, with partial results updated by each iteration. We distinguish each iteration as a separate calculation to get an ordered set Γ. The points of Γ are n-tuples of indices which parameterize the loops with dependences being the (computational) relations from certain lattice points to their neighbours. Thus these dependencies are characterized by a set of vectors v, , ...,V., the components of which are the loop parameters. The resulting structure Ω in physical space-time is also a lattice. The spatial part of Ω is called a systolic array. The most general systolic array (at least in the plane) is the homogeneous extension of the following pattern.
In [5] this lattice, TL is called the universal planner systolic array.
Scheduling Γ in Ω means a mapping of
ν^.,.,ν^Ζ3. 3 2
Since TL — time X TL , this mapping must be such that the spatial components T T T
of image vectors are of the form ±(1,0) ,±(1,1) , ±(1,1) ; these corresponding to the wires of the universal planar systolic array. If this mapping is an isomorphism, the V. must each be at most three dimensional vectors
. That is, we can have at most three loop parameters. Let us consider the example of the matrix multiplication algorithm:
Causality in Algorithms and Architectures 323 DO 10 I = 1 , N
DO 10 e7" = 1 , tf DO 10 K = 1 ,N
A(I, J) = A(I ,</) +B(I, K) * C(K,J)
(6.1)
10 CONTINUE To find the vectors V-^ , V2 , V3 , this program is re-expressed in
so-called pipelined form.
DO 10 I,J,K = 1 ,N
B(I,J9K) =B(I9J-\,K)
C(I,K,J) =C(I-\,K,J)
A(I,J,K) =A(I,J,K-\) +B(I9J,K)*C(I,K,J)
10 CONTINUE From this we can read off the following dependency vectors:
Vl =\ ° />V2 - l · 1 /'V3 =1 ° I' 0 0 , -1
These vectors are the displacement vectors relating index arguments in right and left members of the instructions in the program body. Thus, for example, V2 comes from B(I,JtK) = B(I,J — 1 ,K) .
The theory in [5] tells us that the mapping η , an isomorphism, is a matrix AE GL(3 ,Z) such that the entries of one row of Ad- -£=1,2,3 are less zero (i.e., are time-tike). For instance, an example of the relation
A(dl , d2 , d3) = (Δχ , Δ2 , Δ3)
expressing the mapping η is 1 1 1 \ /-l 0 θ' 0 1 1 ] | 0 -1 0 ,0 0 1 / \ 0 0 - 1
(6.2)
Since time accounts for one row, the resulting array is two dimensional and the spatial part of the dependence may be read off from the framed block in the right member of (6.2). This dependence is specified by the vectors
i° Vo )· O , resp.,
corresponding to dependence on: one datum already present, one datum coming from the left and one datum coming up diagonally from the left, resp., as in
324 W. L. Miranker and A. M. Winkler
the following figure:
D—D-D
This figure contains a portion of the universal systolic array in the plane, and it is indeed the realization of the systolic structure corresponding to the algorithm in (6.1).
Case Study 2: Certain recursion formulas give rise to networks. The Fast Fourier Transform provides a familiar example, with the resulting 'butterfly' network. Peskin [8] has observed a relation between the cyclic reduction algorithm and the ω-network of the NYU Ultra Computer.
The cyclic reduction algorithm solves a tridiagonal system with the generic equation a .x . + b .x . + c .x . — d . by recursive reduction to the diagonal system b1 .x . = d. as follows. From
a . ,x . „ + b . x . + e . x . = d .
a .x . , + b .x . + Q .x . Ί = d . ^ t-l % τ ^ ^ + l ^
a . ,x . + b . x . Ί + e . , , x . i n = d . , . ,
we eliminate x. . and x. , to get
where
a(.2)*. , + ^2 )*.+ Α . = ^2>
/ 0v - a .a . . a ( 2 ) _ ^ ^ - l
^ b . ,
£ i b . . & ·., , ^ - l ΐ, + 1
( 2 ) _ " g t + l g i °i ~ b . Λ τ + 1
,„v d . a . d . o . j ( 2 ) __ , t - l t ^ + l ^ a . — a . r 5 · ^ ^ £ . . 2? . .
We can continue this process, getting
«<*>*. , + Ä.+Ä. + , = d W , fe = 3,4 τ τ-κ % τ % τ + κ ^ ' ' '
Eventually i + k,i—k will exceed the bounds for i, and we will set ^0 = ^ 1
= _2 = · · · = 0 and χτπ+ι=ζχΝ+9 = · · · = 0. Eventually the generic
equation becomes b. x . — d\^ , ^ ^ ^
Causality in Algorithms and Architectures The general recursion of cyclic reduction is c lear ly
(fe+1) a .
~ai-kai
325
b(k+i) = h(k) _ ai-kci _ - - ^ -
τ-k t+k
%
(k) k C . i C .
z+k
,(fc+i) Ak) %-lCi di = di ~ —Jk) °i-k
A<*> d<%™ i+k i b) (k)
ί+k
If we parametize the steps of the algorithm by i and k, we have the impli
c i t dependencies
(i,k) > (i9k + \) , (i-k3k) > i ,k + 1) , (ί + k,k) > (i 9k + \) ,
or equivalently, the explicit dependencies
(i,k) > (i,k + l) , (i,k) > (i + k,k + l) , (i, k) > (i - k9k + l). (The notion of explicit and implicit dependence is discussed in [4].)
This means that Γ has the following form.
i«l i«2 i«3 i*4 i*5 i*6 i«7 i»8
k-l
k*2
k«3
Consider the space-time architecture Ω generated by the following hardware configuration.
D D D D n D a D
BUS CONNECTION
326 W. L. Miranker and A. M. Winkler We abbreviate this hardware configuration with the following diagram
D—D—D—-Q—D—O—D—D
Then Ω, the space-time architecture, which corresponds to Γ, has the following form.
'· Φ—Φ—P—φ—Φ—cp—φ—φ
I I I I I I I I
« ü—ύ—ύ—ΰ—ύ—ö—ό—ύ Here the dashed lines ( ) indicate worldlines, solid lines indicate wires, and we approximate the order relation of Ω by just using time, i.e. (D, 1) > (', 2) > (ü",3) > · · · for any choice of D , D' , · · · . (In these pairs the integer which is the second component is the time.)
We can map Γ ^ Ω (which specifies a schedule of the computation) by superposing the diagrams for Γ and Ω. In Γ, (t , k) > <S' , k1 ) means k<k'
as integers. Moreover under the superposition, k is identified with t, so indeed r\(i, k) > T\(if , k'). Thus η is a consistent schedule.
Note the hardware wiring is recovered by projection of Ω to time £ = 0, as discussed before.
This architecture Ω is not enough to make full use of the parallel structure of Γ, since Γ is naturally structured to admit pipelining. That is, we could keep putting in new values, for each 'cycle1 of Γ, and we seek an architecture Ω which would also allow for pipelined input.
One such architecture is based on the hardware diagram which mirrors the structure of Γ exactly; namely
Causality in Algorithms and Architectures 327 Here all line segments are wires. Then at time 1 we input the first matrix, at time 2 it is moved to the next layer, and we can input the second matrix, etc.
Notice that the resulting network contains the modular ω-network.
00 01 10 II
II 01 10 00
This network is uniquely specified as the inductively defined packet switching network. That is the address (e.g. 01) defines the switching pattern (first left, then right) for passage in either direction through the network. (This property is the one which permits the Fetch & Add operation employed to alleviate signal blocking [2] in packet switching networks.)
Case Study 3: Partitions in a para-tree. A parallel version of a binary search is given by dividing the ordered list <z,<··· < a-, into N segments, where N is the number of processors. Each processor tests the value sought with the middle element of its list, and shares the result of the test with its neighbour. If the value sought is found, the search terminates, and a message is sent to the printer. If not, each processor compares its results with its neighbour's. If each of a pair of neighbouring processors says 'greater than', or each says 'less than', nothing happens. The test results must differ for some pair of neighbouring processors' and the corresponding pieces of the list must contain the element sought. Then correspondingly, we take the right half of the left processor's List, and the left half of the right processor's list as a new list, and repeat the process.
In fact, if all test results agree, then clearly the element which is sought is either in the far left hand processor or the far right hand processor. In this case the modifications required are obvious.
W. L. Miranker and A. M. Winkler
Note that the left and right extremities need only be connected to half the processors, while the others must be connected to all others. Thus we get a complete graph on the interior nodes, which extends out in time in an 'ideal tree' or para-tree.
If the decision process were bounded, we would alternatively obtain the tree architecture being considered for some AI applications [9],
REFERENCES 1. A.J. Bernstein, Analysis of programs for parallel processing, Trans, on
Electronic Computers _, EC-15, (1966). 2. A. Gottlieb, et al, The NYU Ultracomputer—Designing an MIMD shared
memory parallel computer, IEEE Trans, on Computers, C-32, (1983). 3. J.E. Hopcroft and J.D. Ulman, Formal Languages and their Relation to
Automata, Addison Wesley, Reading, Mass., 1969. 4. W.L. Miranker and A. Winkler, Spacetime representation of systolic compu
tational structures, IBM Research Center Report 9775, December 1982. 5. W.L. Miranker and A. Winkler, Spacetime representations of computational
structures, Computing 32 (1984), 93-114. 6. W.L. Miranker and A. Winkler, Causality in Algorithms and Architectures,
IBM Research Center Report 10874, December 1984. 7. D.I. Moldovan, On the design of algorithms for VLSI systolic arrays.
Proc. IEEE 71 (1983). 8. C.S. Peskin, Ultracomputer implementation of odd-even reduction,
Ultracomputer Note #19, CIMS,NYU, January 1981. 9. S.J. Stolfo and D.P. Miranker, A parallel processor for expert systems,
Proc, 1984 Int. Conf. on Parallel Processing (Ed. R.M. Keller), IEEE Comp. Soc. Press.
SOME RESULTS ON FINITE-ELEMENT APPROXIMATION OF A
QUASI-VARIATIONAL INEQUALITY IN SEMI-CONDUCTOR ANALYSIS
N. R. NASSIF Mathematics Department, American University of Beirut,
Beirut, Lebanon
Abstract. This work considers the problem of Galerkin-Finite-element approximation using piecewise linear polynomials for a quasi-variational inequality appearing in semi-conductor modeling. We prove the existence of the Galerkin approximation and obtain estimates in the case of uniqueness of the solution.
1/2 . 1 Specifically we obtain a rate of 0(/z ), in the H -norm.
1. Introduction. The subject of variational and quasi-variational inequalities has recently received a wide attention as it has been found that many engineering problems can be formulated using this approach. Of particular interest, the work of Baiocci in the area of fluid flow through porous media [1], that of Bensoussan, Lions in impulsional control problems [2], and of Hunt-Nassif [3],[4] in the theory of reverse biased semi-conductors.
Let: · U be a Hubert space. • C a closed non-empty convex subset of U9
• a: UXU->R , bilinear, continuous on C, coercive on C — C.
• M: C->U, and one associates with every V € C , the closed convex set K(v) = ψ £ C | ψ < Mv) .
• feu' dual of U . One considers then the problem of finding
u£Ku) : a(u , ψ-w) > (f , ψ-w), ψ € Ku). (1) Joly and Mosco have given sufficient conditions for the existence of a solution [5].
In reverse-biased semi-conductor analysis, these conditions have been verified by Nassif [6] for one-dimensional cases and for some particular situations in two dimensions.
329
330 N. R. Nassif Our interest in this paper is the numerical approximation of (1) using
the finite-element method. Such method has been implemented and yielded some good results in the simulation of semi-conductor devices [7].
2. The Semi-Conductor Model and the Finite^-Element Approximation. Existence of the Galerkin Solution. Typically, in the one-dimensional semiconductor model U= φ£Η10 ,ί) |φ(0) =7_, φ(Ζ) =0; (note that Hl09l)c
C091)). The bilinear form a:U^R is defined by,
*<♦·*> = / i l l ** On U9 using Poincare's inequality we can select the norm ||φ | = α(φ ,φ))£
The operator M is defined as follows. For zEU, W = M(z), solves the boundary value problem:
w" = k[w' ) 2 -w'z' ] , u(0) = V+ , w(l) = 0. (2)
Note that with the change of variables V = e , finding w reduces to solving uniquely the linear boundary value problem
(ekzv')'=0, v(0)=e~ + , v(l)=\. (2'
Note here that -kv
v£V = φ£Η1 0,1) |φ(0) = e + , φ(Ζ) =l ,
w£W = φ£Ηλ (0 , l) |φ(0) = V+ , φ(Ζ) = 0 .
We w i l l use a l s o
W = φ£Η1 ( 0 , Ζ ) | φ ( Ζ ) = 0 .
Let f be a positive constant. In our case (1) takes the form I 1
u€Kz) : f u* φ-u)' dx > \ f<t>-u)dx, \/φΕΚ(ζ), (3) 0 0
where for z £ U
K(z) = (φ€ ϋ\φ < M(z) .
It has been shown in [6], that (3) admits a solution. Let (·>·) denotes the L inner product on (0,1) and ||*||n/ the induced norm.
The finite-element method can be introduced as follows. Let 0 < h < 1, and 7Γ, = x . = ih , 0 < i < N ; Nh = l . Let
#, = φ €# (0 , Ζ-) |φ is linear on (# . , ar . ) , V i .
Define then ^ 7 ^ n 7 , J/- = tf 7 Π W . h h h h h h
For 2 € ί/, , one d e f i n e s ΙΛ s o l u t i o n t o :
( e f c s t > £ , ( φ - υ Λ ) ' ) = 0 , φ € 7 Λ , <2h)
Finite-element Approximation of a Quasi-variational Inequality 331
and W-i is then defined by:
[w'h + v'h/kvh^') =0, <t>ewh . (3h) Henceforth, from well-known results on approximation of elliptic problems W-,
is well-defined as a function of z . We write w-, = MAz) and consider the quasi-variational inequality,
Seek M/z€VM/z): K ' (*-"//)> (ΛΦ"^) , ^ W ' (4h)
where Kh(uh) = φ€£/Λ|φ <«Α(«Α) ·
Let
For z € ίΛ , define £, = 5,2 by the variational-inequality
th<LKhz) : (t'h,V -t[) > (/,φ-ίΑ) Ψ€^(ζ) . t7 is also well-defined from well-known results on existence of solutions of variational inequalities.
LEMMA 1. There exists constants A, B independent of h such that:
z£Ch: \\Mh(z)\\0 <A
\\Sh(z)\\0<B . -kv. kv
PROOF. By letting φ= e + + (l— e + )x/l in (2h), one obtains the first part of the inequality, with:
A = (e +-1) ekd/l d = V+-V_ .
A similar argument to that of lemma 2 in [6] leads to:
\\t'h\f < | |w / -n ' | l o + 2 | | f | | 0 | | ^ | | o , ω = Μ(3)
with u(x) = (l— x/l)d. Using the first part of inequality, one reaches the second part with B , function of A, V+, V_ and ||/ |L.
LEMMA 2. For sGC, with \\z' || < B, and h sufficiently small,
0 <Mh(z) < V+ (i) and
V_ < 5^(2) < V+ .
PROOF. The first part of the lemma is shown by considering the solution v(x) of (e v
arguments that
As v" = —kv' z1', this leads to:
v(x) of (e v' , ψ' — ν' ) = 0 ν ψ Ε 7 . One demonstrates using classical
\v' — vl\\ < oh \\v" \\ , c independent of h .
332 N. R. Nassif
IK-^| | 0<^lb ' ! ! j | * ' | l 0 · 7 1 —fc y
Note that H^' || <e (\-e )/L9 and therefore using the imbedding 1 °°
H (0,l)cC(0,i), one obtains: ||^-^/2ll00< oQh , cQ depending on 7+, 7__, Z- , fe and B.
-kV+ # -fcy Since e < t> < 1 , choosing /z sufficiently small yields e < t;, < 1 . By proper choices of the function ψ in (3h), and since U,> 0 , one demonstrates that wAx .) = - (in[vAx .)])/k, V j . Therefore wAO) = V+ and 0 < wAx .) < 7+. The piecewise-linearity of W-, implies the first part of lemma. For the second part, one has t, < W-, < V+ . Let D,=[x\t,< wA\.
Such region includes x=09 as the smallest limit point. Let x be the other limit point and N-, be the set of piecewise linear functions on (0,x ): let ψ€/1/, such that ψ(θ)= ψ(χ ) =0. For any such ψ one shows that ( ^ , Ψ ' ) = ( / , ψ ) .
Of course ψ is extended to zero outside [0, x ) . This is a piecewise-linear approximation to the problem — y" =f in (0,x )9y(0) = V_ , y[xm) -W-, [x ). Hence, one shows that
max \y-t\ ^ eh \\f\\ , ( 0 9 x ) °
and for /z sufficiently small, t behaves like y which is greater or equal to 7_ on (09x ) . This concludes the second part of our lemma.
Thus by considering the set c°h = i a € < 7 Ä l i a ' l l 0 < » .
the consequence of Lemma 2 is that o o
shtCh*ch> Note that C, is not empty since it contains the approximate Shokley's solution which solves the variational inequality:
uh < V+(\-x/l) : aüh,^-üh) > (/,φ-ΰ^) , φ < V+\-x/l) . We prove now:
LEMMA 3. Let [t, and 3 7 be such that, t-, - SAzA) , and suppose t,-»t, and z-,-*Zi in ίΛ , we then have
( ϋ ) ψ ε χ ^ ) , 3φΒ,εχΛ(8™) such that
limmsup ((**)' , ( φ - φ Ύ ) > 0.
Finite-element Approximation of a Quasi-variational Inequality 333 (i.e. M-, is (a9f) continuous according to [5]).
PROOF, (i) follows from complex compacity arguments contained in [6]. The second one is trivial. Lemmas 1, 2 and 3 are in line with the hypotheses of Joly and Mosco which allow to state.
THEOREM 1. The Galerkin-quasi-variational inequality (4h) admits a solution. For V+ sufficiently small, such solution is unique
3. Error Estimate. It has also been found in [6], that for V+ , sufficiently small, the continuous problem (3) admits a unique solution. This is also true for the Galerkin approximation (4h) . Let u and u-, be such unique solutions. We may write
u < Mu) : (u'h , φ'-w') > ( Λ Φ - u ) > Φ < Mu) , (5)
uh <Mh(uh) :u'hAi-uh)>f,-uh) , φ <Mh(uh) . (5h) Let W=M(u) and w,=MAuA . Letting in (5) φ < u,+ w -w, , we have φ<υ
and therefore [uf ,uf
h -u' + w1 -w£) > f,uh-u + w-wh). (6) Let u-, and W-, be respectively the interpolants of u and W in i/, and W-, .
— h ( τ\\
since u" < 0, then ζλ < u . Furthermore z/ =-^Qe < 0 as cQ > 0 and w" - — kw1 (u1 ~w' ) . It can also be shown when u< w that u1 > — cQ > W1
therefore w11 > 0. Hence W-, > W and u-, < W, . In choosing φ = 2^ + ι^— w^
in (5h) one obtains (6h)
(6) and (6h) yield
Ik-u^Ho < ( ^ - ^ , ^ - ^ ) + ( ^ , ^ - ^ ) + («';,. » ' - ^ )
The second and third inner product on the right hand side cancel as the inter-polant of u is precisely its projection with respect to the inner product (f1>$') · Thus by suitable manipulation one reaches
\\u<-u'n\\2Q < (u'-u'h, w' -w'h) + f,u-uh) + (f,wh-w).
Note now from (3h) and the definition of V and W that
( / / \ (-o1 vh\ Λ n w , c ° ^
Letting φ = ü,—u,, yields.
334 N. R. Nassi f
(w'-yh, u'-u'h) = (W'-w'h,u'~ui) +(-£-ir> u'~u'h hj
, 1 (υ' Vh , ~ Λ
Q-u'-üh) + f i : , W —U
Thus, we may write,
\vh |M' ~ukno < _ i i ü 7 ' M ' _ u Ä J + ^ ' "?>_M^+ (f>uh-w) ih~u-h'
Consider now T the last term on the right hand side of the last and introduce the function V, that solves:
ku7 (e n v'\ = 0 , v(0) = e + , vl) =
From the proof of Lemma 2, we can write:
\\v'-v'h\\Q< &Bh.
Moreover, one shows by a s imple argument t h a t
\\ν'-υ'\\ο< \\v' \Lekd \\u-uh\\0 < C(V+)\\u-uh\\0
with C(V+)->0 as V+ -> 0 . We can w r i t e T a s :
T ~ I V ^ " ' ~ h) *-ΚΪ>~^' h)· T y p i c a l l y
'h v' <e2kV+ \\v(v'h-v') +v/(v-vh)\\Q
2kV„ <e ηιι^-^Ίΐ0 + ^(^)ΐι^-^ιι0]
Using Poincare's unequality, together with (8) gives: \V1 -I
h v < kMh
where M is independent of h. Similarly,
<kE(V+)\\u'-u/h\ 7, V
Introducing the last 2 estimates with (7) gives:
l l" /-^li;<^1ll"-2Äll0+ \\ί\\0\\ν-Ζ*\\η+Μ\\*'-*ί\\ "fc'O hu o + E(V+)\\u'-u^\\l +C2\\u'-u'h h"o'
Finite-element Approximation of a Quasi-variational Inequality 335
For V+ suff ic ient ly small so that E(V+) < 1, th i s can be transformed to :
, * 2 ε , 2 i-E(v+)-M^^]\u' -u'h\\; < cju^^ + | | / · | | 0 ιμ-^ι ι 0
+ f/,2-2ß +C2\\u'-u'h\\Q.
This proves the second result of this paper, specifically, that the order of convergence of the Galerkin approximation is the square-root of the order of the interpolant.
THEOREM 2. In the hypothesis that V+ is near 0, where the original quasi-variational inequality (3) admits a unique solution u, the Galerkin-finite element problem (3h) admits also a unique solution u-, , with the estimate
||u'-«'Ä||0 < < 7 ( ^ + Α 1 - ε + Α) . Note that this estimate is not optimal as in the case of variational inequalities [8].
REFERENCES
1. C. Baiocci, Problemes ä frontiere libre en hydraulique, C.R. Acad. Sei. Paris, 278, series A (1974), 109-112.
2. A. Bensoussan, and J.L. Lions, Inequalities variationelles pour des problemes de controle impulsionnel, C.R. Acad. Sei. Paris, 276, serie A (1973), 1189-1193.
3. C. Hunt, and N.R. Nassif, On a variational inequality and its approximation in the theory of semiconductors, SIAM J. Numer. Anai. , 12 (1975), 938-950.
4. N.R. Nassif, and K. Malla, Une inegalite quasi-variationelle dans la theorie des semi-conducteurs, C.R. Acad. Sei. Paris 294, serie I (1982), 79-83.
5. J.L. Joly and U. Mosco, Resultats d'existence et de regularite pour certaines inequations quosi-variationelles, Lecture Notes in Economies and Mathematical Systems, Springer-Verlag (1975), 107 pages.
6. N.R. Nassif, On the existence and uniqueness for a class of quasi-variational inequalities to solve reverse-biased semi-conductor devices, To appear in Ann. Mat. Pura Appl.
7. N.R. Nassif, and K. Malla, Simulations numerique dTun semi-conducteur polarise en sens inverse au moyen d'une inegalite quasi-variationelle, C.R. Acad. Sei. Paris 294, series I (1982), 345-348.
8. R.S. Falk, Error estimates for the approximation of a class of variational inequalities, Math, of Comp. 28 (1974), 963-971.
ASYNCHRONOUS ALGORITHMS FOR HEAT CONDUCTION IN COMPOSITE MEDIA
M. N. EL-TARAZI and M. N. ANWAR Mathematics Department, Kuwait University,
P.O. Box 5969, 13060 Safat, Kuwait
Abstract. The problem of two dimensional steady-state heat conduction in a medium composed of two homogeneous and isotropic, but physically dissimilar materials is discretized with the method of lines to obtain a system of second order differential equations with multi-point boundary conditions. This differential system is converted, using invariant imbedding for each one dimensional problem, into a fixed point problem and then sequential and parallel asynchronous algorithms are applied.
1. Introduction. Many engineering and physical applications depend on material composed of two or more constituents. Much research has been done and many solutions have been obtained for elasticity and plasticity problems related to such materials. The subject of heat conduction in composite media, however, has received less attention.
In this paper we treat the problem of two dimensional steady-state heat conduction in a medium composed of two homogeneous and isotropic, but physically dissimilar materials. In situations, where the boundary conditions are not simple elementary functions, the exact analytical solution of such problems is impossible even for regions having a simple geometry. Thus a numerical technique is needed and the authors propose in this paper, (as they have done in another paper [l] for Poisson's equation with nonlinear boundary conditions), to discretize the considered heat conduction problem by the method of lines [ll] to obtain a system of second order differential equations with multipoint boundary conditions. This differential system is converted, using invariant imbedding [10] for each one dimensional problem, into a fixed point problem for which several iterative algorithms can be MAA—W
337
338 M. N. El-Tarazi and M. N. Anwar used. If the asynchronous parallel algorithms are used, we obtain then an approach which combines both simplicity (refering to the classical invariant imbedding associated with the by lines approximation) and efficiency (refering to the asynchronous parallel algorithms).
In section 2, we give the mathematical formulation of the problem considered in this paper together with the appropriate boundary and auxiliary conditions. In section 3, we describe briefly the method of lines and invariant imbedding, and its adaptation to the present problem. In section 4, we describe the associated fixed point problem and several sequential and parallel iterative algorithms. In the last section, we apply several sequential algorithms and simulate parallel computation for a numerical example. The results obtained show that parallel algorithms can considerably accelerate classical sequential-type algorithms.
2. Mathematical Formulation. Consider two continua of different thermal conductivities K, and κ~ with an interface S . Let the two domains occupied by the materials be denoted by Ω1 and Ω„ with boundaries S^ and 52
respectively (Figure 1).
FIG. 1. Schematic representation of the domains of the problem
The temperature distributions T-, and Tp in the materials occupying the domains Ω, and Q, respectively, for the steady heat conduction problem, satisfy the Laplace equations (see [8]) given by
t\TY = 0 in Ω (2.1.1)
LT = 0 in Ω2 (2.1 .2)
The problem is supplemented by the appropriate boundary conditions along the boundary S, and 52 . For the sake of simplicity, we assume the temperature to be known along the boundary, that is
τ = τ a long 53
a long 5 3
Asynchronous Algorithms for Heat Conduction 339
T. = Λ, a long 5 . ( 2 . 2 . 1 ) 1 1 1
T2 = A2 along 52 (2.2.2) where A^ and 4 2 are given functions on 5, and S respectively. Additional conditions are needed along the interface £L separating the two domains. These represent the coupling conditions, namely
(2.3.1)
(2.3.2)
where n and n are the outward normals to the subdomains' boundaries. 1 2
It should be mentioned that the boundary conditions (2.2) are of Dirichlet type and other types of boundary conditions such as Neumann or mixed can also be treated by our method. 3. The Method of Lines and Invariant Imbedding. To simplify the exposition of the numerical technique to be used for the solution of this problem, we consider a simpler geometry for the two domains Ω and Ω ; we assume that the composite media is of a cylindrical annulus form. Furthermore we assume no variation in temperature along its axis. Hence, taking a cross section normal to its axis, we obtain three concentric circles S, , <S2 and S?
with radii a 9b and o respectively. Then Ω is the region bounded by the two circles £, and S?, and Ω2 will be the region bounded by the two circles 5 and S as shown in Fig. 2. Therefore, using the polar coordinates v and θ , the relevant equations are
FIG. 2
340 M. N. El-Tarazi and M. N. A n w a r
d2T dT d2T + - — - + -V ö-=0 i n Ω, ( 3 . 1 . 1 )
a / r 3r r2 3Θ2 1
3 2 ? 2 , 3T2 92T2 5 -+ + -ΤΓ Ö = 0 i n Ω0 . ( 3 . 1 . 2 )
dr r dr r2 dQz 2
The boundary c o n d i t i o n s a re T1 = Αχ a long Sx ( 3 . 2 . 1 )
T2 = A2 along S2 . (3.2.2) The interface conditions are
Τχ = T2 along S3 (3.3.1)
BT1 3T2 -5— = - K — — along S , (3.3.2)
where κ = κ /κ and the negative sign is introduced to take into consideration the opposite directions of the outward normals n, and n2 along the interface s3 .
3.1 The method of lines. The method of lines for elliptic and parabolic problems has been described and analysed by G. Meyer [10 , 11], we will adapt it to the above heat conduction problem. Let us introduce the equidistant rays making angles Θ-,, 0 = θ <θ <...< θ = 2π with the horizontal axis and separated by the increment Δθ = Θ, — Θ, for k = 0 , 1 , . . . , n-\ . In the method of lines approximations, we retain one independent variable, say v, continuous, and discretize all the terms involved in the problem with respect to the other variable Θ . If T, 7, and T2 r, denote the solution of (3.1) to (3.3) along the k™ ray, then the method of lines approximation for the problem (3.1) to (3.3) gives, for k— 1 , 2 , . . . , n
II 1 ml , 1 1" 7 + 1 T \ 7 + n (T, 7 1 -2T, 7 + 2\ Vj_A = 0 (3.4.1) i,k r 1, fe ( Γ Δ Θ ) 2 1 yk-l l ,k i,/c+r
T
Τ ι , ί : ( α ) = i4i,fc ( 3 · 4 · 3 )
T 2 , k M = A 2 , k ( 3 · 4 · 4 )
- ^ 1 ^ - « - ^ ^ - ( 3 · 4 · 6 )
The periodicity of the problem implies T = T and T = T . The 1,0 l,n 2,0 2 , n
above system represents a second order two point value problem that can be attacked by a variety of methods. We suggest the application of the method
Asynchronous Algorithms for Heat Conduction 341
of invariant imbedding due to its simplicity.
3.2 The Invariant imbedding. This method (described in detail in [4, 10, 11]) proceeds as follows: the unknown functions T v , T1
v , T v and T'
along the k ray are related through the Riccati transformations
T i , & ] =Ri,kr) y i , f e ( 2 , ) + Wi,kW ( 3 · 5 · 0
T2,k<r)=s2,kM T \ , k ^ + W2,k^- ( 3 · 5 · 2 )
Substituting (3.5.1) [(3.5.2)] into (3.4.1) [(3.4.2)], the problem (3.4) becomes equivalent to the system
ÄU<P>= , + i Ä l , f c W - T ^ T < * ( * > . Ä1>fe(a) = 0 (3.6.,)
^ " ( 2 - Δ Θ ) 2 1 , κ ^ * ( 3 . 6 . 2 )
W 2 fc = ^ T ß2 k^] W2 kr)
' k ( r - Δ Θ ) 2 Z'K , k ( 3 . 6 . 4 )
Rl,k^Ti,k^+Wl,k(C) =F2,k^T2,k(C)+W2,k(a) ( 3 · 6 · 5 )
Γ^ fc(e) = - ^ 2 , / c ( < 3 ) ( 3 · 6 · 6 )
+ ( / i > f e ( r ) ] + r i ) f e + 1 ( r ) = 0 (3.6.7)
+ W2fk(r)] + y 2 j i ; + 1 ( r ) = 0 . (3.6.8)
Equations (3.6.1) and (3.6.2) are integrated forward from r = a to v-b
while equations (3.6.3) and (3.6.4) are integrated backward from r =b to r» = <?. Then we compute the values of T; ^ (ö) and T ·, (c) from equations (3.6.5) and (3.6.6), hence (3.6.7) [(3.6.8)] becomes an initial value problem which is integrated, when an iterative method is used, backward from r=o to v = a [forward from r = o to r = b] . Finally (3.5.1) and (3.5.2) are used to compute T, τ, and T -, respectively.
342 M. N. El-Tarazi and M. N. Anwar
4. Associated Fixed Point Problem and Iterative Algorithms. The approximation by lines problem (3.6.1) to (3.6.8) can be solved by Jacobi, Gauss Seidel or over relaxation type methods. More general algorithms, namely the asynchronous algorithms which allow parallel computation for the problem, can be considered.
4.1 Fixed point problem associated with (3.6). We convert the discretized problem (3.6.1) to (3.6.8) into a fixed point problem of an operator G which will be defined in what follows.
First, for k = l,2,...,n, let
,2 , E2,k = ^ 2 , k e C [°>b] · *2,kW=A2,k
and the product space
(4 .1 )
E = E1,1X XEl,nXE2,l* •xE2,n · ( 4 " 2 )
We now d e f i n e the o p e r a t o r
G : E -* E
(4.3) = K , l ^ ι , η ' v 2 , l ' · · " V2tn]^
[u= ( w l f l , . . . , M l f n , " 2 ) 1 >···> * 2 , η ) where for any /c€l,2,...,n, u 7 and u ν are the solutions in ΕΛ , and E"2 IN respectively to
1 n / \ 2 Ä i , * w - ,+FÄi,*<*> - T ^ J T * U ( r ) . *i,ki°i) - ° <«·*·»
τ > κ (ι»ΔΘ) ' ' (4.4.2)
,2 Äi,fe[Wi.fe-lW ^ t . ^ l W l ' ^ A ' ^ t , * ( Γ Δ Θ ) -
«'ltfc (o) =-K"'2)fe(ö) (4.4.4)
+ ^,fe(^)] + , f e + 1 ( w = 0 (4.4.5)
u. 7,(0) is computed from (4.4.3) and (4.4.4).
and then u . , is computed from »ij f eW=i?.5f e(r)M'.)f e(r)+W.)f e(r), (4.4.6)
where £ = 1 . 2 . a = a and a =b . 1 2
Vfc(e) and W2 ko) , k = 1,2,
Asynchronous Algorithms for Heat Conduction 343 It is clear that u* EE is a solution to the problem (3.6.1) to (3.6.8) if and only if u* is a fixed point to G .
To allow us a greater amount of parallelism (as suggested later), we can uncouple the equations (4.4.3) and (4.4.4) by considering two vectors
. ,n and replacing (4.4.3) and (4.4.4) by
M l , f c ( ° ) = - K [ V k ( e ) " " 2 , ) Ι £ ( β > ] / [ Κ Α 1 , * < β > + Α 2 , * ( " ) ] ( 4 · 4 · 7 )
M2,fe(o) = [Vfc(c) ~ W 2 ,kc)^ ' ^K R i ,ko) + R2,k^o) 1 · <4·4·8) Of course, we have to update f/ .(c) and iv7 iXc) using f/. , and W? , computed from the equation (4.4.2).
4.2 Iterative algorithms. We describe now several iterative sequential and parallel algorithms which we have considered to solve the above problem. Notice first that (4.4.1) has the exact solution
R Ar) - tanh -r-^ Ln — v ' ^ V ΔΘ a
(4.4.9)
where i - 1,2, a and an b . Ί " "2 4.2.1 Sequential Jacobi-type algorithm (J).
We start with the initial guess w = 4 fc = 1,2,...,n. Then For p =0,1,2,... until convergence, do:
For fc=l,2,...,n, do:
" o 7 = A 7 2 ,/c 2 , fe
for
0 2)
3)
use (4.4.2) to compute W -, and W -,
solve the system (4.4.3) and (4.4.4) to compute
L"G(c) and p + i (s )
use (4.4.5) to compute lu^ , J and ί w^ «, )
4) use (4.4.6) to compute (p + 1
1,* and * 2 , *
Of course it is understood that in (4.4.2) to (4.4.6), v, and v^ u and u )
are replaced by vf and t£ (u^ and u^ ) respectively.
4.2.2 Sequential Gauss-Seidel-type algorithm (GS) . This is like the sequential Jacobi-type algorithm except that we make use of the updated vectors as soon as they become available. Of course we can introduce a relaxation parameter ω to obtain a SOR type algorithm.
4.2.3 Modified sequential Jacobi-type algorithm (MJ) . We start with the initial guess u l,k 'ι,λ* U2,K=Al,k> *lV°)= Ml,*(c )· Wl k^)==U2,k0),
344 M. N. El-Tarazi and M. N. Anwar for k = 1,2, . . . , rc . Then For p = 0, 1 ,2, . . . until convergence, do:
For k = 1 ,2, . . . , n, do: 1) use (4.4.2) to compute W, -,
2) use (4.4.7) to compute ir . (c]
3) use (4.4.5) to compute \vr , \
- 4) use (4.4.6) to compute
For k = \ ,2 , . . . ,n, do:
1) use (4.4.2) to compute W, 2,fc 2) use (4.4.8) to compute
3) use (4.4.5) to compute (5:0' - 4) use (4.4.6) to compute u^ τ,
Of course W r,°) [ 9 V^°^ ^s to ^e updated using ίν ^(σ) [ ? ί>(0)] com~ putated from (4.4.2).
4.2.4 Modified sequential Gauss-Seidel-type algorithm (MGS). This is like the modified sequential Jacobi-type algorithm except that we make use of the updated components as soon as they become available.
The above methods are of sequential type. Now we give two algorithms of parallel type designed for multiprocessors.
4.2.5 Asynchronous Jacobi-type algorithm (AJ) . This is an asynchronous parallel version of the modified sequential Jacobi-type algorithm (MJ) defined in 4.2.3; We consider 2a processors P ., j = 1 , 2, . . . ,2a and we assign to each
d one of them the evaluation of n . com-
3 ponents (n + n + . . . 4- n0 = 2n) . 1 2 ^ a Each processor cyclically computes new values of each component in its subset using the values of all other necessary components (in or out of its subset) available at the beginning of a cycle and it releases all updated values at the end of each cycle. To reduce the conflicts among FIG.
Asynchronous Algorithms for Heat Conduction 345 adjacent processors and to simplify the simulation programming, we suppose that the processors are identical and that
nl = n2 = ' ' ' = η2α = n^a (4.4.10) Therefore, each annulus is divided into n/a equal (in number of rays) sectors (Figure 3) and each processor is now in communication with only three others [lower, right (left) and upper], (for the case of two processors we have only side communications).
The 4-steps made by the monoprocessor in 4.2.3 to update one component are exactly carried out in a parallel manner by each processor of the multiprocessor to update one component in its subset.
4.2.6 Asynchronous Gauss-Seidel-type algorithm (AGS) . This is an asynchronous parallel version of the modified sequential Gauss-Seidel-type algorithm (MGS) defined in 4.2.4. It is like the AJ algorithm except that each processor makes use of the updated components in its subset as soon as they become available.
It should be mentioned that the above algorithms are particular cases of the asynchronous algorithms for approximating a solution to a fixed point equation. The asynchronous algorithms were initially introduced by D. Chazan and W.L. Miranker [5] under the name "Chaotic Relaxation" (also called delayed chaotic iterations). The terminology "asynchronous algorithms" was given by G.M. Baudet [2,3] to chaotic relaxation with unbounded delays and he was the first to introduce and experiment on multiprocessor "C. mmp" algorithms like the AJ or AGS on linear algebraic system of equations. Many authors have contributed to the study of the asynchronous algorithms and to related algorithms. A bibliography can be found in M.N. El-Tarazi [6,7] and P. Spiteri [12].
5. Numerical Results and Conclusion. Consider the following heat conduction problem in composite media
Γ 32Γ. , 3Γ. . d2T. — i + I_L + > _ 1 = o, (r,0)en.
Ti(r,Q) = cose + ß^ , (r,Q)£Si
(5.1) < |'2'1 (Ρ,Θ) = Τ2 (Ρ,Θ)
I ( r 9 Q ) e s 3
(Ρ,Θ) = - κ — (ι·,θ) , \ dr 9p
L for i.= 1,2,
346
where
M. N. El-Tarazi and M. N. A n w a r
X = (r,Q)
Ω2 = ( r , 0 )
S =(r,Q)
.S3 =
> , θ )
> , θ )
a <r <o 9 0 < θ < 2 π
c < r < £ , 0 < θ < 2π
r = a ,
r = b ,
0 < θ < 2π
0 < θ < 2π
0 < θ < π
(5.2)
To numerically treat this problem we have implemented the four sequential algorithms J, MJ, GS and MGS on monoprocessor, and simulated the parallel algorithms AJ and AGS (for the simulation schemes we refer to J. Julliand, G.R. Perrin, P. Spiteri [9] and also P. Spiteri [12]) for multiprocessors with 2, 4, 6, 8, 12, 18 and 24 processors.
The numerical computations were made for a = 0.75, b- 1.5, o = 1 , κ = 0.1, 3 =20 and $„=5. We have considered n = 36 rays. Each initial value problem was solved with the classical Runge Kutta method of order 4 and linear interpolation was used when necessary. The step size along each ray was taken as ΔΡ = 1/200. [This means taking 50(100) mesh points in the inner (outer) part of each ray, making a total number of mesh points equal to 36·(50+100) = 5400]. Each iterative process was stopped when
| | u p + 1 - Mp | | < 0 .5X10- 1 · .
Since the exac t a n a l y t i c a l s o l u t i o n of the above example i s known to be
2 \ ( r , θ ) = Μη· Lnr 4- N. + (p . r + — J cos Θ
for £ = 1 , 2 ,
(5.3)
(5.4)
where KM2 = = Μχ =K(^-^2)/Ln [(*)* |
^ — Μχ Lna, N9 = 32 - M2 Lnb
J 6 = £ ( c 2 + a 2 ) - 2 a c 2 + bKa2-cz)
[P2 = 6/[K(c2 + b2)(a2-c2)- (c2-b2)(c2 + a2)]
β2Ρλ = 0 . 5 P2 [ ( C 2 - £ 2 ) - K ( < ? 2 + £ 2 ) ] + ( 1 + K ) £
QY = a l l - a ^ ) , £ 2 = 2 > ( 1 - Z?P2) >
( 5 . 5 . 1 )
( 5 . 5 . 2 )
(5.5.3)
(5.5.4)
(5.5.5)
we could therefore compute the errors of the algorithms considered and found that each one gave numerical results correct to at least 3 decimal-places.
The sequential algorithms J(GS) and MJ(MGS) were almost equivalent. The numerical results are given in Table 1 and in Table 2. The number
appearing after each abbreviation designates the number of processors used. The inner (outer) relaxation is defined to be the set of the basic four
Tabl
e 1
Alg
orit
hm
Inne
r R
elax
atio
ns
Out
er
Rel
axat
ions
Spee
d-up
Eff
icie
ncy
MJ
27 X
36
27 X
36
- —
AJ2
39 X
36
20X
36
2
100%
4J4
2 X
41 X
18
2 X
21 X
18
3.8
95%
AJ6
3 X
41 X
12
3 X
21 X
12
5.7
95%
AJS
4 X
41 X
9
4 X
21 X
9
7.6
95%
AJ\2
6 X
39 X
6
6 X
20
X 6
11 .9
99%
AJ\S
9 X
39 X
4
9 X
20 X
4
17.7
98%
A/2
4
12 X
39 X
3
1 2
X 2
0 X
3
23.4
98%
> o o O
c
TaM
e 2
CQ o
Alg
orit
hm
Inne
r R
elax
atio
ns
Out
er
Rel
axat
ions
Spee
d-up
Eff
icie
ncy
MG
S
16 X
36
16 X
36
- —
AGS2
32X
36
16 X
36
1.5
75%
AGSh
2 X
3.1
X 1
8 2
X 1
6 X
18
3 75%
AGS6
3 X
31 X
12
3 X
16 X
12
4.5
75%
AGSS
4X
31
X9
4X
16
X9
5.9
74%
AGS\
2
6X 2
9 X
6
6X
15
X6
9.4
78%
AGS\
S
9X
31
X4
9X
16
X4
13.1
73%
AGS2
b
12X
31 X
3 12
X 1
6 X
3
17.3
72%
3 X (D
0)
r+ o o g- Ω
rt S"
3 CO
348 M. N. El-Tarazi a n d M. N. A n w a r
actions described in 4.2.1 or 4.2.3 necessary to update one component. The speed-up is the ratio of the execution time of the sequential algorithm MJ (MGS) to the execution time of its parallel version AJ(AGS). The efficiency is the ratio of the speed-up to the number of processors used for the parallel algorithm, and it is given in terms of percentage.
The geometry of the domain (Fig. 3), its uniform partition by the processors and the fact that an outer relaxation costs almost twice as much as an inner relaxation are reflected in the numerical results and they favoured the AJ algorithm in reducing the number of outer relaxations to 20 or to 21 instead of 27 as in the MJ algorithm giving thus an efficiency greater than 95%. These numerical results of parallel computations, obtained by simulation, should only be considered to illustrate the behaviour of parallel algorithms. They show a clear advantage for asynchronous algorithms over synchronous classical sequential methods. We may observe that many different types of parallelism can be considered.
Finally, we conclude by saying that the proposed approach combines the simplicity of the classical invariant imbedding associated with the approximation by lines and the efficiency of the asynchronous algorithms.
REFERENCES 1. M.N. Anwar and M.N. El-Tarazi, Asynchronous algorithms for Poisson's
equation with nonlinear boundary conditions, Computing 34 (1985), 155-168.
2. G.M. Baudet, Asynchronous iterative methods for multiprocessors, Research Report of the Department of Computer Science, Carnegie-Mellon University, Pittsburgh, 1976.
3. G.M. Baudet, The Design and Analysis of Algorithms for Asynchronous Multiprocessors, Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh, 1978.
4. R. Bellman, An Introduction to Invariant Imbedding, Wiley, Interscience, Canada, 1975.
5. D. Chazan and W.L. Miranker, Chaotic relaxation, Linear Algebra and App. 2 (1969), 199-222.
6. M.N. El-Tarazi, These de Doctorat es Sciences, Universite de Besancon, France, 1981.
7. M.N. El-Tarazi, Some convergence results for asynchronous algorithms, Numer. Math. 39 (1982), 325-340.
8. A. Jaworski, Boundary elements for heat conduction in composite media, Appl. Math. Modelling 5 (1981), 45-48.
9. J. Julliand, G.R. Perrin and P. Spiteri, Simulations d'executions paralleles dTalgorithmes de relaxation asynchrone, Rapport de recherche E.R.A., CNRS Micro-systemes et Robotique No. 070906 et de Mathematiques No. 070654, Universite de Besancon, France, 1980.
10. G.H. Meyer, Initial Value Methods for Boundary Value Problems. Theory and Application of Invariant Imbedding, Academic Press, New York, 1973.
11. G.H. Meyer, The method of lines for Poisson's equations with nonlinear or free boundary conditions, Numer. Math. 29 (1978), 329-344.
12. P. Spiteri, These de Doctorat es Sciences^ Universite de Besancon, France, 1984.
LQ-STABLE METHODS FOR CONSTANT COEFFICIENT PARABOLIC EQUATIONS
E. H. TWIZELL* and A. Q. M. KHALIQ** *Department of Mathematics and Statistics, Brunei University,
Uxbridge, Middlesex, UK **Department of Mathematics, University College of Arts, Science
and Education, P.O. Box 1082, Manama, Bahrain
1. Introduction. In recent years, a number of authors have concerned themselves with developing A -stable and LQ-stable methods for the numerical solution of second order parabolic partial differential equations with constant coefficients (see, for example, Lawson and Swayne [5], Lawson and Morris [6] and Gourlay and Morris [l]). The procedure followed by these authors was the one (sometimes referred to as "method of lines") in which the space derivative is approximated by a suitable finite difference replacement and the numerical solution is obtained by solving the resulting system of first order ordinary differential equations. This approach was used by the present authors for first order hyperbolic equations [2,3], for fourth order parabolic equations in [9] and by Twizell [10] for second order hyperbolic equations, and will also be followed in the present paper.
Consider the constant coefficient diffusion equation in one space variable given by
with initial conditions
0 <x <X , t > 0 (1)
u(x,0) = gx) , 0 <x <X (2) and boundary conditions
uQ,t) =u(X9t) - 0, t > 0 . (3)
In (2), g(x) is a given continuous function of x; it is not specified that g(0)=0 and/or g(X)=09 so that discontinuities between initial conditions and boundary conditions may occur and may be propagated as the computation proceeds [6].
The interval 0 <x <X is divided into N+ 1 subintervals each of width
349
350 E. H. Twizell and A. Q. M. Khaliq h so that (N + \)h = X, and the time variable is discretized in steps of length k . The open region R = [0 < x < X] X [t > 0] and its boundary 3i? have thus been covered by a rectangular mesh, the mesh points having coordinates (mh,nk)
with m = 0 , 1 , . . . , N +\ and n = 0 , 1 , 2 , . . . . The notation u = u(mh9nk)
will be used to denote the solution of (1) at the mesh point (mh9nk) while U will be used to denote the theoretical solution of an approximating
m finite difference scheme.
The space derivative in (1) is now replaced by 92u tor h 2ux-h,t) -2u(x,t) +ux + h ,, t) + 0(/z (4)
and (1) , (4) are applied to all N interior mesh points at time t = nk
(n=0,l,2,...). This produces a system of ordinary differential equations of the form
dU(t)
dt = AU(t) , (5)
where U(t) = Unk) = if1 = (U*,Unz , . . . , U?'^)T , T denoting transpose.
In (5) the matrix A is given by "-2 1
A = h~
1
(6)
0 1 -2_ and has eigenvalues λ = — 4/Γ2 sin2[sn72(#+ 1) ] for s = 1 , 2 , · · · , N . A practical difficulty with (5) is that the system is stiff because a large value of N may be required to produce an acceptably small component of the local truncation error relating to the space discretization. This, in turn, leads to a large range of eigenvalues of A and hence to components of the solution with widely varying rates of decay, those components with slower rates of decay affecting the solution longer than those with more rapid rates of decay.
Solving (5) with the initial vector i/(0) = g from (2), gives
Ut) = ex?tA)cf
which satisfies the recurrence relation
U(t + k) = exp (kA) U[t) , t = 0,k92k, . (7)
To obtain numerical solutions from (7), Pade approximants will be used. The (0,1) Pade approximant gives a commonly used four point explicit scheme which
L0-Stable Methods 351
is stable for p < £ , where p = k/h2. The ( 1 ,0) Pade approximant gives the LQ-stable fully implicit or Backward Euler scheme which was extrapolated by Lawson amd Morris [2] and Gourlay and Morris [3] . The (1,1) Pade approximant gives the A- stable Crank-Nicolson method which was also analyzed in [6].
In the following sections of the present paper, higher order Pade approximants will be used in (7) and the resulting algorithms analyzed. The principal part of the local truncation error at the mesh point (mh,nk) may be written down for every method in the form
(_-L tf *±+C fci-liW . (8)
In (8), the component ~~Tjh 9 u/dx is due to the space discretization and the use of (4) in (1); the term C Kr 3^u/3t relates to the approximation to exp(kA) used in (7), the coefficient C being known as the "(time) error
^ 1 constant". For the Crank-Nicolson method q-3 and C~ = - —^ .
J 1 λ
2. A Second Order Method and its Extrapolation. Using the (2,0) Pade approximant to exp(kA) in (7) gives the fully implicit scheme
(I -kA + \ k2A2) U(t + k) = U(t) (9) which is second order accurate in time with C = τ · For (9), the accuracy indicated by (8) is not attained for m=\ or m = N . In a paper on hyperbolic equations, Öliger [7] showed that using lower order approximants near the boundaries does not affect the stability or convergence properties of the scheme as a whole, and the numerical evidence to be reported in Section 5 suggests that this is true for second order parabolic equations also.
Writing z= — k \ , where λ is an eigenvalue of A (so that z > 0) , the amplification symbol of the method given by (9) is
s - ! — 5 - . 1+2+J2
It is obvious that S satisfies |s| < 1 and lim 5 = 0, so that (9) is an LQ-stable method (Lambert [4]).
Applying (9) to each mesh point (mh , nk) , m = 1 , 2 , . . . ,M, at time t= nk leads to a linear system, the unknowns of which are the components of the N -vector U(t + k) . This system is of the form
EU(t + k) +bn ,
where E is a constant matrix of order N ; it has the quindiagonal form
352 E. H. Twizell and A. Q. M. Khaliq
where e, = 1 + 2p + 3p , e2 = -p -2p^
3
e2
(Π)
5^2 + 2p + g p^ . The
s2 ei
e3 e2
vector & is easily seen from (9) to be i/ . The once-only decomposition of # into triangular matrix forms requires
6/17-10 multiplications and divisions and 4/17-7 additions/subtractions. Thereafter, at each application of the method, the evaluation of the solution by forward and backward substitution requires 5/17—6 multiplications/divisions and 4/17—6 additions/subtractions. The method must be applied 12 times in integrating from time t to t+ \2k using a fixed increment k. A count of arithmetic operations shows that the method thus requires a total of 66/17—82 multiplications/divisions and 48/17-72 additions/subtractions to carry out the integration over this particular range of t .
At first sight, this appears unfavourably with the second order Lawson/ Morris method [6] which integrates using a double time step 2k and requires 64/17-40 multiplications/divisions and 44/17-38 additions/subtractions to integrate over the same range of t . However, the error constant for the Lawson/ Morris method is C = | while it is only C3 = g for (9), so that the user who may sacrifice accuracy but must economize fully, can use (9) with a time
2 increment 2k. This increases the error constant for (9) to C~ = τ (still superior to the Lawson/Morris constant) but decreases the total arithmetic operations count to 36/17-46 multiplications/divisions and 28/17-43 additions/ subtractions, both figures representing a considerable saving on the Lawson/ Morris method.
It has been noted already that (9) is L0~stable; it may therefore be extrapolated to improve the accuracy in time. To this end, define
U1) (t + 2k) = ( I -kA + \k2 A2)~2U(t) U (2) (t+2k) = (l-2kA + 2k2 A2)'1 U(t)
(E)
(12)
Then third order accuracy is achieved by U with C =—3. The symbol of
L0-Stable Methods 353
this extrapolated form of (9) is
S 3 V 1 + a + l a 2 j " H l + 2a + 232j
from which it is easy to see that (12) is an LQ-stable method. Implementation of the method requires a total of 114/17-128 multiplications/divisions and 86/17— 1 12 additions/subtractions to integrate from time t to ΐ+12/c.
3. A Third Order Method and its Extrapolation. Third order accuracy in time is achieved when the (2,1) Pade approximant is used in (7) giving
(l-\kA + ik2A2) Ut+k) = (J+ \kA) U(t) . (13)
The error constant associated with (13) is C.=-^- but the accuracy indicated by (8) is not attained for m=\ or m = N.
The symbol is given by . . -*
'*!«*έ«!
from which it is easy to verify that the method is LQ-stable. Applying (13) to each mesh point (mh9nk) , with m = 1 , 2 ,...,/!/, at
time t = nk (n = 0,1,2,...) leads to a linear system of the form (10) . The non-zero elements of the matrix E in (10) are now given by
^Ό + η2 . #„ =-in — 2:n2 . a „ = In . a. = 1 + ^ 7 9 + % Ό1 e 2 = l + 3 P + p , ez=~ip-1p , e3 = ^ p , e^= I + 3 P + IV
Yl and the elements of the vector b by
bn = \pUn + (1 -\p)Un +\ pUnM, for m=\92,...9N m 3 r rn-1 ό ' m 3 ^ m+l
(recall that i/" = U^+l = 0 from (3)). The solution U(t+k) of (13) is determined by applying a quindiagonal
solver. In addition, implementation of the algorithm requires the postmulti-plication of the tridiagonal matrix I+^kA by the vector V_t) . This adds 3/1/-2 multiplications/divisions and 2/17-2 additions/subtractions each time the method is applied, so that the total arithmetic operations count in integrating from time t to time t + \2k becomes 102/17—106 multiplications/ divisions and 76/17-103 additions/subtractions.
In view of its superior time error constant and its lower arithmetic operations counts, the third order method based on the use of the (2,1) Pade approximant in (7) is thus to be preferred to the method based on the extrapolation of the (2,0) Pade approximant.
One of the best third order algorithms to be published within the last five years is that of Gourlay and Morris [3] given by
354 E. H. Twizell and A. Q. M. Khaliq
U t + 3k)=[l-kA) Ut) , U2\t+3k) = [l-2kA) li-kA)'1 Ut) .
U(3\t+3k) = i-SkA)'1 Ut)
This method is L -stable and has C. — ft- . Though not as accurate as (13) it is applied only four times when integration takes place over the interval t to t + \2k and requires a total of 74Λ/-46 multiplications/divisions and 51/V-43 additions/subtractions. The arithmetic operations counts for this method are thus mildly superior to those of (13). However, the error constant (C^ = -=2") of the novel numerical method (13) indicates that it can be used accurately with a time step 3k (thus requiring only four applications to integrate from t to t+12/c). This increases the error constant to C. = Ö (still far superior to that of the Gourlay/Morris method) but decreases the arithmetic operations counts to 38/1/-42 multiplications/divisions and 28/1/-39 additions/subtractions.
The novel third order algorithm (13) can be extrapolated to give fourth order accuracy in time by the sequence of vectors
(l-\kA + \k2A2) V(l) = [l+\kA)V(t) ,
( J - f kA + \k2A2) U(1\t+2k) = (1+ \kA) V(l) , ( 1 5 )
l-\kA + \k2A2)V_2)(t+2k) = (1+ | kA) U(t) ,
The symbol for (15) is
7 V I + 2 a + i
from which it follows that the method is L -stable. The error constant for 8 the method is easily shown to be C' = — »-, c and an arithmetic operations count
shows that 168/1/-164 multiplications/divisions and 128/1/-158 additions/ subtractions are needed to integrate from time t to t+\2k.
4. Two Fourth Order Methods. 4.1 An A^-stable method. Fourth order accuracy is achieved when the (2,2) Pade approximant is used in (7). This gives the algorithm
l-\kA +-^k2A2)U(t + k)= (l+\kA + -~ k2A2) U(t) (16)
for which the error constant is C- = ?n ; the accuracy indicated by (8) is not attained at points adjacent to the boundary where m=\ and m = N .
Applying (16) to each grid point (mh,nk) at time level t = nk
(rn = 1 , 2 , . . . , N ; n = 0,l,2,...) leads once more to a linear system of the
L 0 -Stab le M e t h o d s 355
form ( 1 0 ) . The non-zero e lements of the m a t r i x E i n (10) a r e now given by
TJ1 / I . 5 2 \ rin , / l l \ TFW . i ιΊ1η
-p U* + p ( I _ i p ) ^ + ( l _ p + l p2 ) ^
+ p(?~ip) ^nx +iVp2 i / nx ; ^ = 2 , . . . ,/ι/-ι,
r v z d r/ 7 7 7 + 1 i zr 777+2 -,η l 2 TJn , i l x r.n ,t 5 2x T,n
The solution i/(t + fc) of (16) is computed using a quindiagonal solver and implementation of the algorithm also requires postmultiplication of the quindiagonal matrix I+^kA +— k A by the vector U(t). The total arithmetic operations counts for the method in integrating from time t to time t+\2k are therefore 126/1/-154 multiplications/divisions and 100/1/-151 additions/subtractions.
Clearly, then, the method based on the (2,2) Pade approximant is only minimally more expensive to implement than the lower order methods based on the (2,0) and (2,1) Pade approximants. Unfortunately, examination of the symbol of the method, given by
l-2-*+T2z2
1 +±Ζ + ^rZ 2 2 Ä "" 12 *'
reveals that, while |S|<1 , S-*+l as z -> «> so that (16) is only A -stable.
4.2 An L^-stable method. This method is based on the use of the (2,0) Pade approximant in the recurrence relation (7) and considers linear combinations of the solution obtained over a treble time step 3k . The algorithm
(l) (2) (3) (E) is defined by the sequence of vectors U_ , U , U_ , V_ given by
U(l)(t+3k) = (l-kA + ±k2A2)-3U(t) ,
U2)(t + 3k) = l-2kA + 2k2A2)~1 (l-kA + \k2A2ylV_t) ,
U(3)(t + 3k) = (l-3kA+$k2A2)-1U(t)
U<EHt+3k)=£u™-%Ui2K^U™ . ( 1 7 ) 27 It may be shown that the error constant of this method is C = - -.
symbol is 45 27 _, 2
The
S = 22(1 + z+^z2)6 22(\ +2 z+2zz)(\ +z+±zz) 11(1 + 32 + §22)
from which it is easy to show that the method is LQ-stable; this is its chief advantage over the method given by (16). Counts of the arithmetic operations involved in implementing (17) from time t to t+\2k reveal that, using a basic time step k9 it requires 130Ä/-150 multiplications/
356 E. H. Twizell and A. Q. M. Khaliq divisions and 100/1/-141 additions/subtractions.
A very economic competitor of this novel L - stable algorithm is the fourth order method based on the (1,0) Pade approximant due to Gourlay and Morris [l]. This method is described by the sequence of vectors U_ ' , U , £/<3>, UM, £/<*> where
i/(1)(t+4i:) = 1-kA)-" U(t) , i/(2)(*+4fc) = l-3kA)~l 1-kA)-1 U(t) ,
ü(3)(t+4fe) = (l-2kA)-ll~kA)'2 U(t) , U(h)(t + Uk) = (1-hkA)-1 U(t) ,
U(E)(t + Uk) = 8t/(1) + ^ υ2) -ψυ_(3) -7τυ_Μ . (18)
It may be shown that this method is LQ-stable, has time error constant Cc=~r~ 9 and requires 83/1/-50 multiplications/divisions and 55/1/-18 additions/ subtractions to implement over a time interval of 12k using the basic time increment k.
In view of its far superior error constant, the novel method (17) can be used with a basic time step of 2k thus reducing the number of times it must be applied to integrate from t to t+ \2k from four to two. This, in turn, reduces its arithmetic operation counts to 74717— 90 multiplications/ divisions and 56/1/-81 additions/subtractions; both of these figures are lower than the corresponding figures for the fourth order Gourlay/Morris method. Doubling the basic time step in (17) increases the error constant
Lf. 3 2 . . . . .
to Cs=-5~5~ » which is still superior to the analogous Gourlay/Morris figure by a factor of -~ . The novel method (17), with a basic time step 2k, is thus a more accurate and more economical method than (18). 5. Numerical Results. To illustrate the accuracy and behaviour of some of the numerical techniques discussed in this paper, the heat equation (1) is solved with X=2 and boundary conditions given by (3). The initial conditions are taken to be g(x) = 1 for 0 < x < 2 . This problem was discussed by Lawson and Morris [2] and has theoretical solution given by
oo
ux,t) = V [l ~(-l)S] — sin (£ ST\X) exp (-^S2TT2 t). S=l
The methods given by (9), (13) based on the use of the (2,0) and (2,1) Pade approximants, respectively, in the recurrence relation (7) will be denoted by P20 and P21 and their extrapolated forms by P20E and P21E, respectively. The method based on the (2,2) Pade approximant, given by (16) will be denoted by P22 and the Crank-Nicolson method which is based on the (1,1) Pade approximant will be denoted by Pll; the Lawson/Morris extrapolation algorithm, which extrapolates the (1,0) Pade formulation, will be
Lo-Stable Methods 357 denoted by P10E. The third and fourth order multi-stage methods of Gourlay and Morris [l] will be denoted by GM3 and GM4 respectively, and the novel fourth order multistage method (17) will be denoted by TK4.
All methods are tested using k= 0.025 and h= 0.05 giving p=10, k = 0.\ and h = 0.05 giving p = 40 and k = 0.\ and h = 0.025 giving p=160. The maximum errors at time £=1.2 are given in Table 1.
It is noted from Table 1 that, for the model problem, the most accurate second order method is P20, the most accurate third order method is P21, and the most accurate fourth order method is P21E. These findings are in keeping with the discussions of the local truncation errors of the methods.
Table 1. Numerical Results for the Model Problem
Method
ΡΊ0Ε Pll P20 P20E P21 GM3 P21E P22 TK4 GM4
Sc
Order
2 2 2 3 3 3 4 4 4 4
)lution of
Maxi P= 10 0.48E-3 0.28E-3 0.18E-3 0.74E-4 0.67E-4 0.13E-3 0.60E-4 0.66E-4 0.66E-4 0.17E-4
Lmum error mo p = 40 0.45E-2 0.24 0.17E-2 0.41E-3 0.28E-4 0.17E-2 0.24E-4 0.68E-1 0.16E-3 0.38E-3
differential equation
duli p = 160 0.44E-2 0.52 0.16E-2 0.36E-3 0.22E-4 0.18E-2 0.19E-4 0.30 0.11E-3 0.32E-3
is u(\ ,1.2) s 0.066
It is further noted from Table 1 that, for p=10, the second order method P20 gives better results than the third order method GM3 (Gourlay and Morris [l]) and that the third order methods P20E and P21 give numerical results which are almost as accurate as the fourth order methods. For p = 40 and p = 160 the second order method P20 is seen to be more accurate than the third order Gourlay/Morris method GM3, and the third order method P21 is seen to give better results than the fourth order Gourlay/Morris method GM4.
The 0-stable method P22 gives poor results for the problem, due to the discontinuities between boundary and initial conditions, and behaves in a similar way to the Crank-Nicolson method Pll which was analyzed in Lawson and Morris [6]. It was shown in [6] that, for problems with such discontinuities, oscillations in the solution are induced unless r = k/h<X/i\; a similar
358 E. H. Twizell and A. Q. M. Khaliq analysis for P22 reveals the restriction r < X/π /2 .
The maximum errors given in Table 1 occur at the mid point x = 1 for the L0~stable methods and for the A -stable methods when their oscillation criteria are not violated, and near the boundaries for the A -stable methods when oscillations in the solution are induced.
6. Summary. Second, third and fourth order accurate methods for the numerical solution of the simple diffusion equation in one space variable have been discussed. The methods were compared with a family of methods due to Gourlay and Morris [1],
The favourable results obtained in the numerical experiments showed that the reluctance of some authors (see, for example, Smith et at [8]) to square the matrix arising from the space discretization and the approximation of the space derivative in the partial differential equation, may be dispelled.
REFERENCES
1. A.R. Gourlay and J.LI. Morris, The extrapolation of first order methods for parabolic partial differential equations II, SIAM J. Numer. Anal. 17 (1980), 641-655.
2. A.Q.M. Khaliq and E.H. Twizell, The extrapolation of stable finite difference schemes for first order hyperbolic equations, Int. J. Comip. Math. 11 (1982), 155-167.
3. A.Q.M. Khaliq and E.H. Twizell, Backward difference replacements of the space derivative in first order hyperbolic equations, Comput .-Methods Appl. Mech. Engrg. 43 (1984), 45-56.
4. J.D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley & Sons, Chichester, 1973.
5. J.D. Lawson and D. Swayne, A simple efficient algorithm for the solution of heat conduction problems, Proc. Sixth Manitoba Conference on Numerical Mathematics (1976), 239-250.
6. J.D. Lawson and J.LI. Morris, The extrapolation of first order methods for parabolic partial differential equations I , SIAM J. Numer. Anal. 15 (1978), 1212-1224.
7. J. Öliger, Fourth order difference methods for the initial boundary value problem for hyperbolic equations, Math. Com. 28 (1974), 15-22.
8. I.M. Smith, J.L. Siemieniuch and I. Gladwell, Evaluation of Norsett methods for integrating differential equation in time, Internat. J. Numer. Analyt. Methods in Geomech. 1 (1977), 57-74.
9. E.H. Twizell and A.Q.M. Khaliq, A difference scheme with high accuracy in time for fourth order parabolic equations, Comput. Methods Appl. Mech. Engrg. 41 (1983), 91-104.
10. E.H. Twizell, An explicit difference method for the wave equation with extended stability range, BIT. 19(3) (1979), 378-383.
FINITE ELEMENT CALCULATION OF A PATH INTEGRAL FOR NONLINEAR
FRACTUREt
J. R. WHITEMAN* and G. M. THOMPSON** * Institute of Computational Mathematics, Department of Mathematics and Statistics, Brunei University, Uxbridge,
Middlesex, UK **GKN Technological Centre, Design and Analysis Group,
Wolverhampton, UK
1. Introduction. Path independent integrals for use in fracture mechanics were presented by Eshelby [4] and Rice [6], The original integral of this type, the J-integral, was proposed for problems of infinitesimal elasticity. However, in recent years a large literature generalising this integral to elastic contexts has been created; see, e.g. Atluri [l], Blackburn [2], Blackburn et al [3], Hellen [5], and Rice [7].
In the present paper the finite element method is applied to nonlinear fracture problems in which elastic-plastic deformation with hardening takes place. The elastic-plastic behaviour is modelled incrementally, see Whiteman and Thompson [9], and finite element approximations to a modified form J
of the J-integral are derived. It should be noted that the J -integral was in fact derived using deformation theory of plasticity, thus embodying the assumption that no unloading occurs as deformation takes place. Numerical values for JO are presented for three two-dimensional crack problems.
2. Jp-integral. Rice [6] proved that, for a homogeneous two-dimensional elastic body, the rate of decrease in potential energy with respect to crack length is equal to the path independent integral, J. That is
dPE J = , (2.1)
dL where Pp is the potential energy and L the crack length. J is defined for a crack with flat surfaces parallel to the x -axis, see Fig. 1, as
du . 1
_ - Π Wdx0 - T. — — ds 2 Ί dx-
(2.2)
Invited talk. 359
360 J. R. Whiteman and G. M. Thompson where the contour Γ surrounding the crack tip, starts from the lower crack surface and continues in an anticlockwise direction until it ends on the upper surface, W is the strain energy density, u- are the displacements, and the tractions T · are defined with respect to the unit outward normal vector n . so that Tj = o . .n . .
1 tj J
crack surfaces
FIG. 1. Two-dimensional crack with contour Γ
For problems of linear elastic fracture the value of J can be used as a fracture criterion. In the present work for elastic-plastic fracture problems the e/p-integral is used similarly. This integral can be developed from the J-integral (2.2), by separating what was for the elastic case the strain energy density W into elastic and plastic components, W^ and f/ respects y tively. Thus
W = We + W , (2.3)
where the elastic component is given in terms of the stress and elastic strain components by
^ = Κ·,·(εν*)Λ· (2.« ^ΰ ^ΰ e The plastic term, W.', is defined as_
wpsf'° d V (2.5)
where σ and ε are respectively the effective stress and effective plastic strain. Equations (2.2) —(2.5) together define J .
3. Calculation of the J -integral. Full details of the method of approximating Jp throughout the load history for elastic-plastic deformation are given by Whiteman and Thompson [9]; a description of the algorithm and use of the MODEL finite element code appears in [8].
The finite element method is applied to problems involving cracks using a formulation in terms of displacements. With incremental plasticity the load is applied incrementally, and for the k™ increment the finite
Finite Element Calculation of a Path Integral 361 element approximation du^ to the corresponding increment of displacement is first calculated. The associated increments of strain and stress, de (*)
.<*> <fc) and
(k) do^ are then retrieved and the total displacements u\ , strains ε_-(k) Z '"
and stresses σ, , for the combined load up to the fctn increment, are calculated.
For the approximation of J-, as in (2.2) —(2.5), to be calculated, a contour path Γ, see Fig. 2, is chosen through a ring of elements üe sur-
(k) rounding the crack tip. The total value of the approximation (j )) is e (k) P
calculated by summing the contributions (j ), from elements involving seg-ments Γ of Γ. Each element Ω is transformed in turn onto a standard
element in the (ξ1 , ξ2)-ρΐ3ηβ so that the image of Γ is either a line ξ1 = Const, a line ξ = Const or a corner path consisting of parts ξ. = Const and ξ2 = Const.
FIG. 2 . The contour Γ around the crack tip, point 0. The Gauss points are indicated by X and Γ by the dotted line. The upper crack surface is the line 0A .
(e) Along the image of Γ , for example ξ. = Const, we have that
v p'h _ V Ae) (Ai) Aj)\(k) r (3.1)
(Ί) (7) where the (ζ^ » ξ« ) a r e Gauss points and the G. are the corresponding weights. If the stresses and displacements at the end of the /cth load
ι i Ak) , , es(k) . . . 0 increment are respectively [O..), and \u.)-, , ^,t7=l,z, M), Ak) . το η τ n
then
ITA- ')
* < · > <
is given by
(k) = , e Ak) d^Ulh , e dx
(k)fduih d(u
,e Ak) d ^ h (k) 02ih dXn
+ WP
dx2
dx2
^7
eAk) 2h dx,
362 J. R. Whiteman and G. M. Thompson
3(ζΛ( ? 0 / / e >(fc) _,, (e)>(fe) 1 yu\'h , i.e »(fe) . , e * (fc) \
9 ^ Bx \2 /dx \ 2 1 1 / 2
(3.2) [©+ mi
= \ ä™de™ J P
where f/ is the plastic work term at the end of the k i increment given p
by e<fe)
0 For nonlinear fracture problems it is normal to choose a number of contours
(k) Γ and to calculate a mean value for (J )7 . In this way for a problem P . (k)
involving a total of L load increments an approximation (^O)h
k = 1 ,2 , . . . ,L , is calculated for each load increment, so that a sequence of values is produced over the entire loading history. 4. Application to Crack Problems with Strain Hardening. The crack problems analysed are shown in Fig. 3. They are (a) single-edge crack specimen, (b) double-edge crack specimen, (c) centre crack specimen. The mesh for the half-body or quarter-body is also shown in Fig. 3. Each is a Mode I crack problem with an applied stress of 100.0 and an initial uniaxial yield stress of 100.0. Plane strain is assumed and Young's modulus and Poisson's ratio are taken respectively to be 1.25 E4 and 0.25. Each problem is solved assuming successively linear elasticity, linear elasticity-perfect plasticity and power law strain hardening. Eight node isoparametric quadratic elements are used, and are taken in collapsed form at the crack tip. Quarter-point crack tip elements, see [9], are employed for the linear elastic analysis but not for the other cases. The power law for the strain hardening is taken to be
σ ( öY
where ε and σ are respectively the effective plastic strain and effective stress, σ is the uniaxial yield stress and a and n are hardening parameters. In (4.1) a is set to be 1.0, whilst n is taken successively as 0.1, 1.0, 3.0, 5.0, 7.0, 10.0. Six load increments are chosen for problem (a), whilst five increments are taken for problems (b) and (c).
3.2
3.2
Finite Element Calculation of a Path Integral
(a) (b)
363
0.4
0.8 1.6
(a) Single-edge crack problem (b) Double-edge crack problem (c) Centre crack problem
1.6
J •
1
t _i
1 0.4 0.4
FIG. 3. The three Mode I nonlinear fracture problems with finite element mesh
364 J. R. Whiteman and G. M. Thompson (a) J results for the single-edge specimens
^ « v l n c r e m e n t ^ssvNumber
n ^ X .
L.E.
0.1 1.0 3.0 5.0 7.0
10.0 P .P .
1
7.542
0.07328 0.07328 0.07328 0.07328 0.07328 0.07328 0.07328
2
-
0.3742 0.3669 0.3648 0.3652 0.3665 0.3658 0.4178
3
-
1 .721 1.615 1.639 1.672 1.693 1 .712 3.674
4
-
5
-
4.203 7.484 4.055 7.801 4.456 10.78 4.788 13.49 5.000 15.13 5.186 1 17.29
D I V E R G I
6
-
11.72 12.89 24.90 36.65 44.33 61.40
N G
(b) J results for the double-edge specimens
^ s l n c r e m e n t ^ χ Ν umber
n v ^
L.E.
0.1 1.0 3.0 5.0 7.0
10.0 P . P .
1
12.79
0.07307 0.07307 0.07307 0.07307 0.07307 0.07307 0.07307
2
-
0.2193 0.2197 0.2186 0.2184 0.2182 0.2181 0.2263
3
-
0.4950 0.4994 0.5015 0.4997 0.4986 0.4982 0.5406
4
-
1 .203 0.9398 0.9376 0.9386 0.9395 0.9416 2.201
5
-
2.080 1 .665 1.496 1.495 1 .499 1.502
10.14
(c) Jp results for the centre cracked specimens
P r e i n c r e m e n t ^ \ N u m b e r
n ^ v .
L.E.
0.1 1.0 3.0 5.0 7.0
10.0 P . P .
1
1 .305
0.05730 0.05730 0.05730 0.05730 0.05730 0.05730 0.05730
2
-
0.2297 0.2308 0.2274 0.2272 0.2271 0.2270 0.2441
3
-
0.5910 0.5423 0.5360 0.5350 0.5345 0.5344 0.9125
4
-
1 .410 1 .090 1.022 1.012 1 .012 1.013
5
-
2.102 1.909 1.780 1.756 1.764 1.771
D I V E R G I N G
FIG. 4. The mean Jp results for the three fracture problems. L.E. and P.P. denote respectively linear elastic and elastic-perfectly plastic analyses
Finite Element Calculation of a Path Integral 365
The results are displayed in Fig. 4. For each problem, the mean value (k) of (Jj)i is given after each load increment for each hardening rule. In
each case, the value of (« D)t increases with the load factor. For the single-edge and centre cracked problems there is divergence during the perfectly plasticity analysis. The value of the approximation initially decreases with n but as the load factor increases, it increases with n.
The increase oi crack problem.
(k) The increase of (Jp)^ with n is particularly marked for the single-edge
REFERENCES
S.N. Atluri, Path-independent integrals in finite elasticity and inelasticity, with body forces, inertia and arbitrary crack face conditions, Eng. Fracture Mech. 16 (1982) 341-364. W.S. Blackburn, Path independent integrals to predict onset of crack instability in an elastic plastic material, Internat. J. Fracture 8 (1972). 343-356. W.S. Blackburn, T.K. Hellen and A.D. Jackson, An integral associated with the state of a crack tip in a non-elastic material, Internat. J. Fracture 13 (1977), 183-200. J.D. Eshelby, The continuum theory of lattice defects, Solid State Physics, Vol.Ill, pp.79-144, Academic Press, New York, 1956. T.K. Hellen, Numerical methods in fracture mechanics, of G.C. Chell (ed.), Developments in Fracture Mechanics — 1, pp.145-181, Applied Science, London, 1981. J.R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35 (1968), 379-386. J.R. Rice, Elastic-plastic fracture mechanics, of F. Erdogan (ed.), pp.23-54, The Mechanics of Fracture, ASME-AMD 19, ASME, New York, 1976. G.M. Thompson and J.R. Whiteman, The use of the MODEL finite element code in the solution of problems of linear elastic and nonlinear fracture, Technical Report BICOM 84/4, Institute of Computational Mathematics, Brunei University, 1984. J.R. Whiteman and G.M. Thompson, Finite element calculations of parameters for singularities in problems of fracture, of J.R. Whiteman (ed.) pp. 27-47, The Mathematics of Finite Elements and Applications V, MAFELAP 1984, Academic Press, London, 1985.
NUMERICAL TREATMENT OF SOME MATHEMATICAL MODELS DESCRIBING
LONG-RANGE TRANSPORT OF AIR POLLUTIONt
Z. ZLATEV* Air Pollution Laboratory, Riso National Laboratory,
Danish Agency of Environmental Protection, DK-4000 Roskilde, Denmark
Abstract. A fairly general class of mathematical models for studying the long-range transport of air pollutants is presented. Each model within this class is described by a system of partial differential equations (PDE1s) . The numerical treatment of such systems of PDE1 s is discussed from a general point of view. It is shown that it is worthwhile to apply a splitting algorithm by which the system of PDE1s describing the mathematical model is decomposed into three parts. These parts correspond to the components of the physical phenomenon studied:
(i) transport caused by the wind, (ii) diffusion of pollutants in the atmosphere, (iii) emission sources and sinks (together with chemical reactions). The first part obtained in the splitting process is very important (the
wind is the main reason for the Long-range transport) and, therefore, it must be handled in a very careful way. The application of variable stepsize variable formula methods (VSVFM1s) in the time-integration of the system of PDE1 s found in the first part is discussed. It is illustrated that if the programming of the numerical algorithms is correctly done, then the variation of the time-stepsize follows the variation of a certain norm of the wind velocity vector. This leads to a very efficient numerical handling of the problems under consideration. Moreover, it is demonstrated that the results obtained are very reliable, which is absolutely necessary because the results have to be used in different studies by other specialists (first and foremost by biologists).
This work has been supported in part by grants from the Council of the Ministers of the Nordic Countries and from the EMEP (European Monitoring and Evaluation Program) project.
"f" Invited talk. 367
368 Z. Zlatev 1. Description of a General Air Pollution Model. The long-range transport of pollutants in the atmosphere is often described by a system of q partial differential equations (PDE ' s) of the form:
3Cs *°* 3°s ,*"* + 3 (y ^sV 9 3 C ^^ δ 3 ΰ^ + β π n
= 1,2, where
(i) <? is the number of different pollutants involved in the physical process under consideration,
(ii) c = cs (x ,y ,z tt) , s = 1,2, ...,<?, is the concentration of the s pollutant in the atmosphere,
(iii) u = u (x9y , z , t ), V = V (x,y , z , t) and W=w(x,y,z,t) are wind velocities along the Ox , <9z/ and Oz axes,
(iv) Κχ= Kx(x,y9z9t) , Ky =Ky(x,y ,z,t) and Z^ = (a?, 2/, z , t) are diffusion coefficients,
and (v) Qs = QQ (x,y ,z,t ,cl ,c2 , . . . ,σ ), s = 1 ,2, . . . , q , represents
the emission sources and/or sinks for the stn pollutant (chemical reactions are often incorporated in these terms).
The space domain, in which the system of PDE's (1.1) is defined, is normally rather simple. In many cases the parallelepiped:
def V = x,y ,z)/x £ [a1,b1], y£[a2,b2]9 ze[a3,b3] (1 .2 )
can successfully be used (see, for example, Zlatev [35]). If the terrain is taken into account and the resulting domain, say V , is not a parallelepiped, then a domain V, which is parallelepiped, can still be obtained by performing an appropriate mapping
F;P*-P; (1.3) see, for example. [26], [9], [4] or [23]. This means that the assumption that the space domain is defined as in (1.2) is not a restriction for the class of problems under consideration.
Some boundary and initial conditions have to be attached to (1.1). These conditions depend on the particular pollutants studied and may change when different pollutants are considered. Therefore, for the purposes in this paper where the general model (1.1) is studied from a numerical point of view, it is sufficient to assume that some appropriate boundary and initial conditions are in use. Nevertheless, it must be stressed that one should be careful with the boundary conditions; in some cases the boundary conditions may affect the performance of the numerical algorithms.
Numerical Treatment of some Mathematical Models 369 The first three terms in the right-hand side of (1.1) describe the
adveotion transport. When long-range transport of air pollutants is studied, these terms have a very great influence on the solution; the influence of these terms is normally much greater than the influence of the other terms. Moreover, the first two of these three terms, which describe the horizontal transport, are more important than the third term, which describes the transport in the vertical direction. There are several reasons for this. The intervals [α,,Ζ?,] and [a2,&2] are rather long; their lengths are normally several thousand kilometres (see [36]). The length of the third interval is rather small in comparison with the first two; b~-a^ is as a rule not larger than 3 km. This implies that the increments Δ# and ky used in the space discretization are much larger than the increment Δ2 . Thus, the horizontal sides of the grid-boxes with edges Δχ, Δζ/ and Δ2 must necessarily be much larger than the vertical sides (in order to reduce the computational work). Therefore some mixing of the concentrations considered within the grid-boxes has to be assumed. If this is done, then the vertical transport is not very important even when the values of W are quite large. Therefore it is not a surprise that the vertical transport is omitted in same models for studying the long-range transport of pollutants in the atmosphere.
The next three terms in the right-hand side of (1.1) describe the diffus
sion phenomena in the atomsphere. The most important of these three terms is the third one, which describes the diffusion in the vertical direction. K (x ,y, z ,t) is often considered as a piecewise constant in a z function. A special function h(x ,y ,£), which is called the mixing height in meteorology, may be used in the definition of Kz(x ,y ,z ,t) . As an illustration only, let us define Kz(x ,y ,z ,t) as
Kzx,y9z,t) =0 if h(x,y,t)<z<b3, (1.4a) Kz(x9y,z,t) =K*x,y,t) if a^ < z <hx , y , t) , (1.4b)
where Kz(x 9y,t) is a given function. It should be mentioned here that function h(x ,y ,t) takes positive values only.
The fact that the diffusion phenomena are not so important as the advec-tion phenomena when the long-range transport of air pollutants is studied explains why many modellers assume that Κχ and K are constants and define K as in (1.4). Sometimes the diffusion in the horizontal directions is totally neglected in the models. For the general treatment in this paper no special assumption concerning the diffusion terms is needed. However, the particular results, which will be presented in the last section, were obtained by the use of constant X and K.. and by assuming that K„ is
x y ά MAA—Y
370 Z. Zlatev determined by (1.4) (these assumptions were also used in [40], [41]).
The last terms in the right-hand side of (1.1), the terms Qs s = 1,2, . . . , q) , describe the sources and the sinks in the system as well as some chemical reactions between the different pollutants involved in the model. The complexity of these terms depend both on the particular pollutants that are considered and on the chemical reactions that are taken into account in the model. Normally these terms are rather complicated (see. for example, [23]). However, in some special but very important cases, as the study of sulphur pollution or the study of nitrogen pollution, only simple chemical reactions between the pollutants involved in the model may be taken into account. In this way the model may be simplified considerably and, what is more important, it becomes possible to handle the model numerically on long time-intervals when simple chemical reactions only are taken into account.
From the above discussion it becomes apparent that the terms in the right-hand side of (1.1) describe three different physical phenomena:
(i) advection transport (caused by the wind), (ii) diffusion of the pollutants in the atmosphere,
and (iü) emission sources and sinks together with some chemical reactions.
When system (1.1) is considered in connection with the long-range transport of a group of air pollutants, the advection phenomena play as a rule the most important role.
2. Splitting of the General Air Pollution Model. From the physical point of view it is important to choose the parameters in model (1.1) (especially those involved in the terms Q , s= 1 , 2, . . . , q) in a proper way. In this paper it is assumed that such a choice is already made and some problems arising in the numerical treatment of model (1.1) will be studied. In order to reduce the computational work it is worthwhile to split the model. Very often the multidimensional models (in our case, the three dimensional models) are split to several one-dimensional models ([2], [21], [23], [30]). However, one could also split the model so that each of the parts obtained after the splitting describes one of the three physical processes listed at the end of Section 1. Such a splitting could be defined as follows:
9c* de* de* do* -TTJT = —u -K v -ς w -z— , s = l,2,...,(7; (2.1) dt dx dy dz ^
-τΓ = τϊΚχ ^ + fc [Ky ~W) + ^ [Kz -Si",)· S - 1.2,...,<?; (2.2)
3c*** — % j r = Q a , B = 1,2, . . . , q . (2.3)
Numerical Treatment of some Mathematical Models 371 Consider the grids KM= xm/xl=al> Xm<Xm+l ( * = l , 2 , . . . , t f - l ) , x^b^Men, (2 .4 )
vN= yn/y1 = a2, yn<yn+1 (n = 1 , 2 , . . . ,N- 1), yN = b2, New] , (2.5)
ZP= ^Zp/zl=CL39 Ζρ<Ζρ+ι (P= 1 , 2 , . . . , P - 1), 2 p = &3, PGJN. (2.6)
Denote
G = X X L X Ζπ . (2 .7) M N P
Note that no requirement for equidistant grids is imposed in (2.4) — (2.7). However, the fact is that very often the real life computations are carried out by the use of equidistant grids in the space discretization. If this is so, then the increments along the coordinate axes are
Δχ = b1-a1)/M , (2.8)
Ay = (b2-a2)/N , (2.9)
As = (b3-a3)/P · (2.10) The results, which will be presented in the last section, are obtained
by the use of equidistant grids; however, such an assumption is not needed for the general discussion in this section.
Assume that the space derivatives are discretized in some way (at the grid-points of G) . By the use of the values of the space derivatives obtained during the space discretization and the values of the known functions at the grid-points of G the systems of PDE's (2.1) —(2.3) could be reduced to three large systems of ordinary differential equations (ODE's):
^ = ft,g*) , (2.11)
4£- - f**(t,g**) , (2.12) at
f*** (t9g***). (2.13) dt
In the above systems ^ 6 ^ " " , ^** E E " " " and g*** € ~RJ L X L if^^^LXL _ j _*** s-^LXL·
vectors whose components are values of the concentrations (of all q pollutants), while f*e]RL X L, f**eB.LXL and f *** € ]RL X L are the discretized right-hand sides of (2.1), (2.2) and (2.3) respectively. This means that
L = qMNP (2.14)
and it is clear that L is normally very large; its order is often 0(10 ). Therefore it is important to choose efficient numerical algorithms for solving (2.11)—(2.13). One should especially be careful when system (2.11) is solved. This is so not only because, as mentioned in the previous section,
372 Z. Zlatev the influence of the advection terms on the solution of (1.1) is very great, but also because the time-stepsize that is to be used in the numerical algorithm depends on the variation of the wind velocity vector. Because of the stability requirements the time-stepsize could, in principle at least, be large when the norm of the wind velocity vector is small, while the time-stepsize must be small when the norm of the wind velocity vector is large; see Zlatev [35], [36], Zlatev et at [37], [38], [39], [40], [41]. Monitoring the norm of the wind velocity indicates that it varies rather quickly and in rather large intervals; an illustration is given by the upper curve in Fig. 2.1 .
The above discussion and the behaviour of the wind velocity shown on Fig. 2.1 indicate that it is worthwhile to allow the time-stepsize to follow the variation of the norm of the wind velocity vector in order to reduce the number of time-steps in the numerical integration of (2.11) and, thus, to reduce the amount of the computational work. This means that the following strategy is worthwhile and should be very efficient if correctly implemented.
Strategy 2.1 Pa/it 1 — Attempt to de^tgn a method ^οκ solving (2.77) In
which, the. timc-btcpbtze. chosen at each time.-6te.p ΙΛ αλ lange. <u pobbible., but
back that both the. accu&acy and the. stability a/ui ph.e.i>eA.\je.d duJving the. whole.
lnte.gnatißn pnoce&i>. Pcuvt 2 — Adjust the. methods LU>e.d In the. solution o^
[2.12) and 2.13 t>o that the.y axe. tn borne, -oejooe compatible, with the. method
cho6e.n in the eolation o^ (2.11).
The second part of the strategy is very important. It is not sufficient to solve efficiently the system (2.11) only. Fortunately, in many cases it is rather straightforward to adjust the methods used in the solution of (2.12) and (2.13) so that the global solution process (in which the method selected for the solution of (2.11) is incorporated) is accurate and stable, so that the results obtained are reliable. This is true for the cases where sulphur pollution and nitrogen pollution are studied (and this will be demonstrated by numerical results in the last section). However, in some cases, especially when complicated chemical reactions are included in (2.13), the adjustment of the numerical methods used in the solution of (2.12) and (2.13) to the numerical method used in the solution of (2.11) could be a very difficult task.
In the next section the problem of choosing a numerical method for solving (2.11) will be discussed.
Numerical Treatment of some Mathematical Models 373
THE VARIATION OF THE WIND VELOCITY VECTOR AM) THE TIME STEPStZ
THE MONTH IS DECEMBER 1982
fW ΓΛ/\/\/^> 1 1 1 1 1 1 r
0 6 10 15 20 2S 50 40
THE UPPER CURVE THE LOWER CURVE
THE TIME AXIS (THE TIME 18 MEASURED IN 0AY8) THE VARIATION OF THE NORM OF THE VINO VELOCIT THE VARIATION OF THE TIΗΕ-βΤΕΡβIZE
FIG. 2.1 Comparison of the variations of the norm of the wind velocity vector and the time-stepsize. The norm of the wind velocity vector is measured in m/sec., originally the time-stensize was measured in minutes, but on the figure all time-stepsizes" are divided by a factor of 6 in order to separate the two curves
374 Z. Zlatev 3. Efficient Numerical Treatment of the Advection Part of the General Model.
In the previous section it has been shown that the solution of system (2.11), arising after the space discretization of the advection terms in 0 · 1) is a very important part of the global solution process. It has also been shown that the time-stepsize should be. allowed to follow a certain norm of the wind velocity vector in order to reduce the computational work. A greater flexibility could be achieved if the time-integration algorithm is also allowed to vary from one time-step to another. This indicates that the use of variable stepsize variable formula methods (VSVFM* s; see Zlatev [31], [32], [33], [34], [35]) is profitable for this class of problems. However, the implementation of a VSVFM is not an easy task. Some of the problems arising in the implementation of VSVFM1 s for solving the particular system (2.11) will be discussed in this section.
Rewrite (2.11) as ^ = f(t,g). (3.0
Assume that F is a set of s numerical integration algorithms. An arbitrary element
F. e F d i f \F.IJ e I , 2 , . . . , S d e = f M (3.2)
i s given by μ[.0] v t 0 ]
40]= Σ<#ν*+Δ* I # V < i - \ i=0
J o ] . [o]v fk = f^kgk ) s (3.4)
3 3_
g[r] = V a[r)gv . + At ß [ r ] / [ r " l ] +Δ* ) ' ß[r) f. , (3.5) i=l i~\
r = \ , 2 , . . . , s .
f[kP] = f(tk,9l
kr]) , P = 1 . 2 , . . . , e j . . (3 .6 )
Index j is only used to indicate that the formulae (3.3) —(3.6) represent element F. € F. Writing (3.3) —(3.6) as above an assumption is made that a constant time-stepsize is applied. Let
. —(MM'M11 lmJ.W). <3.» If the first s . - 1 approximations (g , g , . . . , g Ä _ ) of the solu-
0 i z s j 1 tion of (3.1) are already calculated in some way, then the computations with
Numerical Treatment of some Mathematical Models 375 (3.3) — (3.6) could successfully be carried out for k = s* , s* + 1 K ,
J ΰ where K is such that (b-a)/K= Δ£. At the end of the computations at each time-step k , g-, and j\ are set equal to g-, 0 and /*, 3 respectively. This shows that (3.3) —(3.6) is a predictor-corrector (PC) scheme in which the prediction performed by (3.3) is followed by s- corrections (3.5). If
[0] J (3.3) —(3.6) is used as a PC scheme, then 3— = 0 should be assumed. The use of the predictor formula only (S. = 0 A 3 - ° -0) will also be allowed. The implicit formulae used in the numerical treatment of the so-called stiff ODE's could also be found as a special case of (3.3) —(3.6) by taking S. = 0 A 3 - =£ 0 . This means that a fairly general class of numerical methods J J for solving systems of ODE's is covered by (3.3) —(3.6).
If only one scheme, say F., of set F is in use in the manner described J
above, then it is said that a constant stepsize constant formula method (CSCFM) is applied. We are interested in the transition from CSCFM's to VSVFM's. This transition could be carried out as follows. Introduce the non-equidistant grid:
GK = V*0 = a ' tk>tk_l(k=\929...,K)itK=b, ΚΖΈ^ . (3.8) Let At, = t-, - tj._ λ be the time-stepsize used at time-step k where
fc= 1,2,...,#. Then F · takes the form μί·°] ' ν ί ° ]
40]= Σ $1(A'w> H-i+ Σ Δ ν . 4°] K;> h-i. (3.9) i = 1 i = 1
f[0]= fit a[0]) Jfe nzk'gk * ' (3.10)
yr ]
a? =
r =
κ,-1 , 2 ,
1 , 2 , ,
>ί . . . .
> . . ,
> - i ]
s . ,
J
,l>] V .
J
♦I <Z>1
( 3 . 1 1 ) J
/ "= f ( t ^^ r ] ) , r = 1,2 e. . (3.12)
Now the coefficients are not constants as in (3.3) and (3.5). They depend on the last s* time-stepsizes used in the integration process or, in other words, on the components of vector At-, . defined by
*1· = <Δνι / ΔνΔν2 / Δν · · ·>Δν3*+ι / Δν · (3· ,3)
A relationship between (3.3)-(3.6) and (3.9)-(3. 12) must be established. This can be done (Zlatev [33], [34]) by imposing the requirements:
376 Z. Zlatev (i) The scheme defined by (3.9) —(3.12) must be of the same order as the scheme defined by (3.3) —(3.6) for all vectors Δ£, . .
* J
(ii) If all components of vector btv . are equal to 1 (or, in other words, ΰ . *
if the same time-stepsize has been used during the last s. time-steps), then (3.9)-(3.12) must reduce to (3.3)-(3.6). DEFINITION 1. If the above two conditions are satisfied, then it is said that (3.9)—(3.12) is a scheme corresponding to scheme F. € F with regard to vector bit-, . . DEFINITION 2. Let t,£G* for Vfc . If at every time-step k the scheme (3.9)—(3.12) is correspondent to some element of set F with regard to the current vector Δ£7 . , then it is said that the elements of F are basic kj schemes for the VSVFM under consideration.
Let us try now to investigate how the fundamental properties (consistency, zero-stability and convergence) of the CSCFM's are preserved in the transition to VSVFM's. It is well-known that there are simple algebraic conditions for consistency and zero-stability ([6], [7], [8], [9], [14], [16], [20], [29]). Moreover, it is well-known that consistency+ zero-stability =*· convergence ([29]). It is desirable to find a class of CSCFM's such that if a set F of basic schemes is selected from this class, then the VSVFM's
containing schemes that are corresponding to the basic schemes of F are consistent, zero-stable and convergent when the basic schemes of F are consistent, zero-stable and convergent. In this way the fundamental properties of the constant stepsize constant formula methods which are used as basic schemes in F are automatically transferred to the VSVFM 's induced by the schemes in F. Some restrictions on the time-stepsize selection strategy are needed in order to achieve such an automatic transfer. DEFINITION 3. If the relations
Δ£* = max (At.), kt*K<c < <*> (i.e. if_ X->0, then VAt, -» 0 ) , (3.14) \<k<K K
0 < S < Ltklbtkmml < 3 < °° for Vk£ 1 ,2, . . . 9κ) d=f K* , (3.15)
are satisfied, then it is said that a stable stepsize selection strategy is used.
The subclass of the class defined by (3.3) —(3.9), for which an automatic transition of the consistency, zero-stability and convergence from CSCFM's to VSVFM1 s can be proved, is defined as follows.
DEFINITION 4. The basic schemes in set F are called two-ordinate if
Numerical Treatment of some Mathematical Models 377
[r] [r] [r] [r] [r]
s = 3 , 4 , . . . , y p = 0 , 1 , 2 , . . . , s . , j € M , d J
and if the order of the predictor is V . , while the order of the 2»th corral 3
rector is vL. + 1 ( r = 1 , 2 , . . . , s .) . 0 J
It is obvious that the two-ordinate basic schemes are consistent, zero-stable and convergent (see, for example, [14]). Assume that in the schemes (3.9) —(3.12) that are corresponding to the two-ordinate basic schemes of F the coefficients 3 . .(At* .) (3 = 0 , 1 , . . . , s ; i = 0, 1 , . . . , v ; fc=l,2,
^ Λ [0] 3 . . . ,K) are used to obtain the required order (v. for the predictor and [r] 3
V. + 1 for the correctors). It can be proved that it is always possible V
to achieve such orders by the use of these coefficients ([33] , [34]). Moreover, if this is done, then
aljj (AtJ\) = a[F.] (ι: = 1,2,...,μ^; r= 0 , 1 , . . . , s . ; j € M ; k € K*)
(3.17) is satisfied ([33],[34]). Therefore
|a^](At^.) (j=l,2...,s; r = 0 , l 8 .; i = 0, 1,.. ., ^.; k=l,2,...,tf)
are called /ree parameters.
DEFINITION 5. If the scheme (3.9)-(3.12) used at any step k € K* is such that s* < k, then the VSVFM is called self starting.
d
The following theorem ensures the transition of the fundamental properties (consistency, zero-stability and convergence) from CSCFM's to VSVFM1 s when the basic schemes of set F are two-ordinate. THEOREM 1. Assume that the following conditions are satisfied : (i) the basic schemes in set F are two-ordinate, (ii) the sets of coefficients
a^](At* J)(i = 0,l>...,pJ; r = 0 , 1 , . .., 0j. ; j € M , fc £ K*) contain free parameters,
(iii) the stepsize selection strategy is stable, (iv) the VSVFM is selfstarting,
[s.] (v) 0 < α. ΰ < 2 , Vj €M .
V
Then the VSVFM is consistent, zero-stable and convergent.
This result could be generalized ([34] , [35]), but it is sufficient for the purposes in this paper.
The above theorem ensures that if the stepsizes used are sufficiently small, then one should expect the results to be accurate. Moreover, the
378 Z. Zlatev errors will not blow up when the stepsizes are decreased. This is an excellent property of the numerical algorithms, which is absolutely necessary, but it is not sufficient. In practice, one is as a rule interested in achieving accurate results by using large stepsizes. This could be achieved if the numerical methods used are both sufficiently accurate (of high order) and absolutely stable when large stepsizes are applied in the computational process. In order to explain the requirement for absolute stability, let us consider the simple partial differential equation
3c 0 o -— =— u -— (u being a constant) . (3.18) at ox
Assume that pseudospectral (Fourier) discretization (see [19] , [24]) of ÖO/ÖX is applied on a space grid containing 2M+ 1 equidistant grid-points. Then a system of ordinary differential equations
d-f--usg t e n » 1 . « « ' * ^ ' * 1 ) ) , ( 3 · 1 9 )
dt y \ / in which the components of vector g are values of the unknown function a(x,t) at the space grid and S is a skew-symmetric matrix induced by the discretization operator, can be obtained. System (3.19) is an analogue, concerning the semi-discretization of (3.18), of the famous test-equation dg/dt = — ^g ([8]). It can be proved (see, for example, [38]) that if the time-stepsize satisfies
/h. Lt<(JS^ Χ ) Δ ϊ , (3.20)
\|ΐί|π M-\)
then the computations with the time-integration algorithm under consideration (used as a CSCFM) are absolutely stable. The parameter h. is the
imag length of the absolute stability interval on the positive part of the imaginary axis for the time-integration algorithm under consideration ([38]).
Criteria similar to (3.20) could be formulated for more general systems ([35] , [36] , [40]). One could not guarantee stable computations when such criteria are used in connection with more complicated systems of PDE1 s and/ or in connection with VSVFM's, but there are good reasons (based on heuristics) to expect absolutely stable results when such generalized criteria are applied. The expectation is enhanced considerably if the stability check (based on some of this criteria) is combined by an accuracy check ([35],[40]).
All stability criteria (similar to (3.20)) used in connection with semi-discretizations in which only first-order space derivatives are involved depend on h. . Therefore it is important to derive time-integration schemes for which this parameter is as large as possible. However, by the following theorem it is established that h. could not exceed certain
imag
Numerical Treatment of some Mathematical Models 379 limits. The theorem is deduced from a more general result proved in [18].
THEOREM 2. The interval of absolute stability on the positive part of the imaginary ; limited by imaginary axis for schemes of type (3.3) —(3.6) with s . correctors is
0 < h. < s .+ 1 . (3.21) imag j
Since it is absolutely necessary to preserve the fundamental properties (consistency, zero-stability and convergence) of the schemes (3.3)—(3.6) in the transition from CSCFM's to VSVFM 1 s, two-ordinate basic schemes must be used in set F. Theorem 2 together with the discussion given before this theorem indicate that one should try to determine the sets of free para-meters or. (j = 1 , 2 , . . . , s ; r = 0 , 1 , . . ., s .) in the two-parameter schemes (3.3) —(3.6) so that each basic scheme has as large as possible ^imag· A search for basic schemes with large h. has been carried out by the use of some subroutines from the HARWELL Subroutine Library. The basic schemes obtained in this search were used in the calculation of the results that will be presented in the last section. The use of such schemes ensures consistent, zero-stable and convergent results (because they are two-ordinate; see Theorem 1). Moreover, since h. is close to the upper bound in (3.21),
' imag the use of these schemes combined with some stability and accuracy criteria allows the code to select large stepsizes when the norm of the wind velocity vector is small and vice versa. This means that the computational process is nearly optimal (in the sense that the use of large stepsizes is allowed if this is possible). Thus the theorems given in this section allow us to optimize the computation. The optimization should be understood as follows: if the space discretization is performed in some way and if the time integration algorithms used in connection with systems (2.12) and (2.13) are fixed, then the time-integration of (2.11) could be carried out in a nearly optimal way by the use of a VSVFM based on two-ordinate schemes (3.3) —(3.6) with large parameters h. . The efficiency of this procedure will be demonstrated in the last section. Several remarks are necessary before the numerical illustrations. REMARK 1. In general, it will not be difficult to select time-integration schemes for solving (2.12) and (2.13). It is clear that in the stability criteria h . (the length of the absolute stability interval on the nega-
real tive part of the real axis) should be used instead of ^ m a g when (2.12) is solved. Schemes with large absolute stability regions (and even implicit methods) are needed in the solution of (2.13). Some of the methods discussed in [17] could be applied in the solution of (2.12) and (2.13).
380 Z. Zlatev REMARK 2. Methods based on linear multistep formulae have been considered in Section 3. Other methods could also be applied; methods of Runge-Kutta type, for example. If methods of Runge-Kutta type ([1], [3], [16]) are in use, then it is easy to preserve the fundamental properties in the transition from CSCFM's to VSVFM's. However, these methods require many function evaluations (evaluations of the right-hand side of the system) and, therefore, are rather expensive for the class of problems under consideration (this has been verified experimentally in [39]).
REMARK 3. The result given in Theorem 1 covers perhaps the most general class of time-integration algorithms for which the fundamental properties (consistency, zero-stability and convergence) are ensured without very restrictive assumptions on the stepsize selection strategy and/or formula selection strategy. If some extra assumptions concerning the stepsize selection strategy and/or the formula selection strategy are imposed, then results similar to those in Theorem 1 could be proved for more general classes of methods. Results concerning the fundamental properties of the VSVFM1 s are given in [5], [11], [12], [131, [22], [25], [28].
4. Numerical Results. Some numerical results obtained with a simplified version of model (1.1) will be presented in this section. This simplified version of the model has been used to study the long-range transport of sulphur and nitrogen pollution over Europe. Results concerning the long-range transport of sulphur pollutants will be discussed. In this particular case the general model (1.1) is simplified because: (i) the use of q = 2 is sufficient (the pollutants involved in the process
being sulphur dioxide, S02, and sulphate, SO^); (ii) the vertical transport could be omitted (which is formally achieved by
setting W= 0) ; (iii) K and K could be considered as positive constants;
x y .
(iv) K could be defined as a piecewise constant in z function by (1.4); (v) the use of very simple chemical reactions is quite sufficient.
The particular form of the simplified version of (1.1) obtained by the use of the above assumptions could be found in [40] , [41] . In this paper we are interested in the answers of the following questions: (A) Is it efficient to use the VSVFM's when the long-range transport of
sulphur pollutants over Europe is studied? (B) Are the results obtained by the use of a VSVFM, in which large and
even very large time-stepsizes are normally in use, reliable ? (C) If the answer to the previous question is positive, how can this fact
be exploited ? The answers to these questions are sketched below.
Numerical Treatment of some Mathematical Models 381 (A) The ability of the code to select time-stepsizes according to the variation of the norm of the wind velocity vector (large stepsizes when this norm is small and vice versa) is clearly demonstrated in Fig. 2.1. The relationship between the norm of the wind velocity vector (the upper curve) and the time-stepsize (the lower curve) is precisely as it should be. The code achieves such a perfect relationship by carrying out both a stability check and an accuracy check at the end of each time-step (see also [35] , [36], and [40]), (B) The reliability of the results obtained by the model has been checked by comparing them with measurements. Such comparisons have been carried out on yearly, seasonal and monthly bases. The results of the comparisons are represented both graphically and statistically on more than 100 figures and tables in [42] . The comparisons show that the calculated (by the model) concentrations and the measurements are in good agreement. A very good agreement between calculated and measured concentrations has also been achieved in the study of the long-range transport of nitrogen pollutants over Europe. (C) The fact that the model performed very satisfactorily in the comparisons (carried out in [42]) could be used to evaluate concentrations due to emission sources in a given country. Such an experiment, where the country is Denmark, is discussed in [42]. The calculations indicate that only about 30% of the sulphur pollution in Denmark is due to Danish emission sources; the other part is caused by the emission sources that are located outside the country.
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3. K. Burrage, J.C. Butcher and F.H. Chipman, An implementation of singly-implicit Runge-Kutta methods, BIT, 20 (1980), 326-340.
4. T.L. Clark, A small-scale dynamic model using a terrain following coordinate transformation, J. Comput. Phys. 24 (1975), 186-215.
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7. G. Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Trans. Roy. Inst. Techn. No.130, Stockholm, 1959.
8. G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 27-43.
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382 Z. Zlatev
10. C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1971.
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12. C.W. Gear and D.S. Watanabe, Stability and convergence of variable multistep methods, SIAM J. Numer. Anal. 11 (1974), 1044-1053.
13. R.D. Grigorieff, Stabilität von Mehrschrittverfahren auf variablen Gittern, Preprint Reihe Mathematik Nr, 89, Techische Universität Berlin, Berlin, 1981.
14. p. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley, New York, 1962.
15. A.C. Hindmarsh, LSODE and LSODI, two new initial value ordinary differential equation solvers, ACM SIGNUM Newsletter 15 (1980), 10-11.
16. P.J. van der Houwen, Construction of Integration Formulas for Initial Value Problems, North Holland, Amsterdam, 1977.
17. P.J. van der Houwen and B.P. Sommejer, Predictor-corrector methods with improved absolute stability regions, IMA J. Numer. Anal. 3 (1983), 417-437.
18. R. Jeltsch and 0. Nevanlinna, Stability of explicit time-discretizations for solving initial value problems, Numer. Math. 37 (1981), 61-91.
19. H.O. Kreiss and J. Öliger, Comparison of accurate methods for integration of hyperbolic equations, Tellus 24 (1972), 195-215.
20. F.T. Krogh, VODQ/SVDQ/DVOQ-Variable Order Integrators for the Numerical Solution of Ordinary Differential Equations, Report, Jet Propulsion Laboratory, Pasadena, California, USA, 1969.
21. G.I. Marchuk, Methods of Numerical Mathematics, Springer, Berlin, 1975. 22. R. März, Variable Multistep Methods, Preprint No. 7, (Neue Folge)
Sektion Mathematik, Humboldt-Universität zu Berlin, Berlin, DDR, 1981. 23. G.J. McRae, W.R. Goodin and J.H. Seinfeld, Numerical solution of the
atmospheric diffusion equation for chemically reacting flows, J. Comput. Phys. 45 (1982), 356-397.
24. S.A. Orszag, Numerical simulation of incompressible flows within simple boundaries: Accuracy, J. Fluid Mech. 49 (1971), 75-112.
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stepsize variable formula methods, Numer. Math. 37 (1981), 157-166. 33. Z. Zlatev, Consistency and convergence of general linear multistep
variable stepsize variable formula methods, Computing 31 (1983), 47-67. 34. Z. Zlatev, Variable Stepsize Variable Formula Methods Based on Predictor-
corrector Schemes, Report No, 261, Danish Center for Applied Mathematics and Mechanics, Technical University of Denmark, DK-2800 Lyngby, Denmark, 1983 (to appear in Applied Numerical Mathematics, 1 (1985)).
Numerical Treatment of some Mathematical Models 383
35. Z. Zlatev, Application of predictor-corrector schemes with several correctors in solving air pollution problems, BIT 24 (1984), 700-714.
36. Z. Zlatev, Mathematical model for studying the sulphur pollution over Europe, J. Comput. Appl. Math. 12 (1985), 651-666.
37. Z. Zlatev, R. Berkowicz and L.P. Prahm, Choice of time-integration scheme in pseudospectral algorithm for advection equations, In Computational Methods for Fluid Dynamics (K.W. Morton and M.J. Baines, eds.), 303-321, Academic Press, London-New York, 1982.
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41. Z. Zlatev, R. Berkowicz and L.P. Prahm, Package ADM for studying long-range transport of pollutants in the atmosphere, _Tn: PDF Software: Modules, Interfaces and Systems (B. Engquist and T. Smedsaas, eds.), 153-169, North-Holland, Amsterdam, 1984.
42. Z. Zlatev, R. Berkowicz and L.P. Prahm, Studying the sulphur pollution over Europe, Report No. MST LUFT-A98, Air Pollution Laboratory, Danish Agency of Environmental Protection, Riscf) National Laboratory, DK-4000 Roskilde, Denmark, 1985.
FUNCTIONS MOYENNE-PERIO DI QUES ASSOCIEES A UN OPERATEUR
DIFFERENTIEL DANS LE DOMAINE COMPLEXE*
K. T. MECHE Department de Mathematiques, Campus Universitaire Belvedere,
1060 Tunis, Tunisie
Ab s tract: We denote by JC (C) the space of the entire functions in the complex plane, and by J^ '(C) its topological dual.
We consider the differential operator of order m, m ^ 1, in C:
m k L = Σ a (z)D , k = 0 k
where D = -7-, a, G #' (C) , k = 0, 1 , . . . , m - 1 ; a = 1 . dz k m
In this paper we prove a representation theorem for the generalized mean-periodic functions of ,Κ' (C), and study the expansion in series of the generalized mean-periodic functions of JC (C), relative to an analytic functional of J^ f(C).
Introduct ion
On des igne par:
- fid (C) l'espace des functions entieres sur C muni de la topologie de la convergence uniforme sur tout compact de C. C'est un sous espace ferme de ξ(ΙΙ12) (espace des functions de
oo Λ
clas se C sur IR ) .
* Invited talk. MAA—Z
385
386 Κ. Τ. Meche - Ji?1 (C) le dual topologique de Jf (C) . Les elements de
J^(C) sont appeles fonetionnelles analytiques. D'apres le theoreme de Hahn-Banach toute fonetionnel1e analytique se pro-longe en une forme lineaire continue sur ξ(ΐΙ12).
On considere l'operateur differential d'ordre m, m 2 1, sur C:
m L = Σ a.(z)DJ
j=0 J
ou D = -r-, a . e J T ( C ) , j = 0 , 1, 2, ..., m - 1; a = 1. dz j m
Dans ce travail on etablit un theoreme de representation des fonctions moyenne-periodiques generalisees de >tf (C) et on etudie le developpement en serie des fonctions de .tf (C) qui sont moyenne-periodiques generalisees par rapport ä une fonetionnel1e analytique de JT!(C).
Pour les fonctions moyenne-periodiques classiques dans le domaine complexe, des resultats analogues ä ceux ci ont ete demontre par L. Schwartz, A.O. Gelfond, A.F. Leont'ev (voir [7] [3] [4]).
Le plan de ce travail est le suivant:
On donne dans le premier paragraphe les resultats de J. Delsarte et J.L. Lions [2] sur les Operateurs de transmutation de 1T
Operateur L en l'operateur D sur Jif(C).
A l'aide de ces Operateurs on definit dans le premier paragraphe le produit de convolution generalise, et on etablit un theoreme de Paley-Wiener pour la transformation de Fourier-Borel generalised associee ä l'operateur L.
Dans le deuxieme paragraphe on definit les fonctions moyenne-periodiques generalisees de .Xf (C) , et on donne un theoreme de representation de ces fonctions.
A la fin de ce paragraphe on montre quTil y a identite entre les fonctions moyenne-per iodiques generalisees de Jif(C) et les
Fonctions Moyenne-perio di Ques Associees
f o n c t i o n s f de Jt? (C) v e r i f i a n t :
387
r L ( f ) * j i r (c )
ou f (f) est le sous espace ferme de JC (C) engendre par les Yf quand Y parcourt le groupe des automorphismes Y de J6 (C) tels que :
YL - LY
On donne dans le troisieme paragraphe le developpement en serie des fonctions de Ί((C) qui sont moyenne-periodiques generalisees par rapport a une fonetionnelle analytique de J^f(C).
I. Transformation de Fourier-Borel et Produit de Convolution associes a l'Operateur L.
1) Operateurs de transmutation.
On considere l'operateur differentiel d'ordre m, m £ 1, sur C:
m L = Σ a.(z) DJ
j-0 J
ou D = -£-, a. e JT(C), j = 0, 1, 2, ..., m - 1; a = 1 dz' j ' J ' ' ' m
On designe par ψ, la solution de l'equation:
L ψ (z) = λ ψ, (z), λ e C
ψχ (0) = 1, D^ ψλ(0) = XJ, 1 ύ j ύ m - 1
Cette solution est une fonction entiere en λ et
On considere le developpement en serie entiere de la fonction λ - ψ χ ( ζ ) :
Vz ) " n Vz> £ n = 0
Les fonctions Θ , n G IN, qui appartiennent ä # (C), sont
388 Κ. Τ. M e c h e
c a r a c t e r i s l e s p a r l e s r e l a t i o n s :
I L Θ = 0 , s i n < m-1 ; D J 0 ( 0 ) = j ! 6 . , j = 0 , 1 , . . . , m-1 n ' n J n , j J
L Θ = T — ^ - K - r Θ s i n > m ; D J 0 ( 0 ) = 0 , j = 0 , 1 , . . . , m - 1 . n ( n - m ) ! n-m n
THEOREME 1.: ([2]) p. 666 - 691: L'operateur X defini par:
x ( f ) ( Z ) = r D ^ ; 0 ) Θ (z ) n = 0 n !
est un automorphisme de J^(C) verifiant:
LX(f) = X D m f , f G ^(C)
V (f)(0) = DJf(0), j = 0,1,. . . ,m-1
L'operateur inverse de l'operateur X est donne par:
°° m-1 DPLkf (0) p + km X_ 1(f)(z) = Σ Σ (p+km)! z
k=0 p=0
REMARQUES 1) L'operateur Xest appele Operateur de transmutation de l'operateur L en l'operateur D sur .Xf(C).
2. On a les relations
- V z E C , V λ G C, ψχ(ζ) =X(e X')(z)
- V n G IK, V z G C, θλ(ζ) - X(c m ) (z)
2) Transformation de Fourier-Borel generalisee et theoreme de
Paley-Wiener associes a IrOperateur L.
On designe par Exp(C) l'espace des fonctions de Jf (C) de type exponentiel on a:
Exp(C) = U Exp (C) R>0 R
Fonctions Moyenne-perio di Ques Associees 389
Exp_(C) = f e JT(C)/N(f) = sup|f (λ) |e"RlXl < + «> K λ C
L'espace Expn(C) muni de la norme N(f) est un espace de Banach.
On muni t l'espace Exp (C) de la topologie de limite inductive.
DEFINITION 1. La transformee de Fourier-Borel generalised d'une fonctionnelle analytique S de tf f (C) est la fonction de Jf (C) definie par:
f(S)(Ä) = < S ^ X > .
La proposition suivante donne les relations formation et la transformation de Fourier
^ 0(s)Q) - <s,eX*>, s e .*· (c) .
PROPOSITION 1.; On a les relations
Vs e JT(c) ,^(s) =^ 0 0 ^(s)
oii lest 1 ' automorph isme de J^(C) defini par:
<ÜX(S) ,g> = <S, X(g)>, g G .#(C) .
THEOREME 2. (Theoreme de Paley-Winer) : La transformation de Fourier-Borel generalisee & est un isomorphisme d'espace vectoriels topologiques de Ji?1 (C) sur Exp (C).
Demonstration
Le resultat decoule de la Proposition 1? du Theoreme 9.2 de [8] p 475 et du Theoreme 1.2.
qui lient la trans--Borel ^ n definie par:
3 ) Produit de convolution generalise assooie a I ' Operateur L.
390 Κ. Τ. Meche Du Theoreme 1. et du the"oreme de Cauchy-Kowalewska on deduit le resultat suivant:
PROPOSITION 2. Soit f une fonction de .tf (C). Le probleme de Cauchy:
(O L u(z,w) = L u(z,w) z w
DJ u(0,w) = ^ J f ( w ) , j = 0,1,...,m-1
<£s = X D J X J r-i
admet une solution unique dans J<f(C) donnee par;
u(z,w) =X zDC w[X- 1(f)(z + w)]
DEFINITION 2. on appelle Operateurs de translation generalisee assoc ies c ,tf(C) par: associes a l'operateur L, les Operateurs T , z G C, definis sur
V w G C, T f(w) = X X [X_1(f)(z+w)] z z w
REMARQUE: La definition des Operateurs de translation generalisee associes ä un Operateur differentiel dans le domaine complexe, comme etant l'unique solution d'un probleme de Cauchy du type (1.1); se trouve dans le livre de B.W. Levitan [5] p 104-118.
PROPOSITION 3. Les Operateurs T verifient les proprietes suivantes:
i) Pour tout z G C, l'operateur T est lineaire continu ' r z de ,W (C) dans lui meme.
ii) La fonction z > T est entiere z
iii) Pour toute fonction f dans .W(C) on a: T f(w) = T f(z) ; Tnf(z) = f(z) z w u
T 0 T f = T O T f ; L T f = T L f z w w z z z
i v ) On l a f o r m u l e de p r o d u i t
Fonctions Moyenne-perio di Ques Associees 391
Τζψχ(ν7) = ψχ(ζ) l|^(w) .
DEFINITION 3.
1) Le produit de convolution ge*ne*ralise de deux fonetionnelles analytiques τ et S de Jif(C), est la functionnelle analytique T#S de JT'(C) de*finie par:
<T#S,f> = < Tr,<S ,T ί(ζ)>> , f e JT(C).
2) Soient S une fonetionne1le analytique de ^'(C) et f une fonction de Jf^(C). Le produit de convolution generalise de S et f est la fonction de Ji?(C) definie par:
S // f(z) = <S r, T ί(ζ)>. s» z
REMARQUE: Corame l'operateur de translation generalisee associe ä D est l'operateur de translation habituelle, par consequent le produit de convolution generalise associe a D d'une fonction-nelle analytique S de Jif'CC) et d'une fonction f de ,tf (C) est le produit de convolution habituel defini par:
S*f(z) = <SC, f(z+C)>.
PROPOSITION 4.
1) Soient τ et S deux fonctionnelles analytiques de ^'(C) et f une fonction de .#(C), on a:
x#(S#f) = (x#S)#f.
2) Soient τ et S deux fonetionnelles analytiques de ^ ' ( C ) , on a:
THEOREME 3. Soit S une fonetionnelle analytique de ^ ! ( C ) et f une fonction de Ji? (C) . On a les relations suivantes:
(lX ~HS) # X(f) = X(S*f) ZX(S) * X_1(f) = X_1(S//f)
392 Κ. Τ. Meche 11. Fonctions Moyenne-periodiques General isles et Groupe g ^_
1) Fonctions moyenne-periodiques generalisees.
DEFINITION 4. On dit qu'une fonction f de Jf(C) est moyenne-periodique generalisee
, si le sous espace ferme V (f) engendre
par les T f, z G C, verifie: z
vL(f) * .#(c)
D'apres le theoreme de Hahn-Banach cette definition est equi-valente a la definition suivante:
DEFINITION 5. On dit qu'une fonction f de JT(C) est moyenne-periodique generalisee s'il existe une fonetionnelle analytique S de jrf(C), non nulle teile que:
S // f(z) = 0, pour tout z G C
EXEMPLES.
1) Les fonctions moyenne-periodiques generalisees associees ä l'operateur D sont les fonctions moyenne-periodiques classiques, car on a:
V (f) = V_(f) Dm D
ou V (f) est le sous espace ferme de Ji?(C) engendre par ί(ζ+ζ), z G C.
2) Soit zQ G C, z * 0. Les fonctions f de ^ ( C ) verifiant:
T f(zQ) = 0, pour tout z G C
sont moyenne-periodiques generalisees, car on a:
T f ( z . ) = 6 , # f ( z ) z 0 z0
ou 6 est la masse de Dirac au point z„. z 0 o
Fonctions Moyenne-perio di Ques Associees 393
3) La fonction:
V*(z) = (^T ν ζ ) ) / μ - λ
est moyenne-periodique generalisee, car on peut trouver une fonetionnelle analytique S de Jif'(C), non nulle, teile que sa transformee de Fourier-Borel generalisee ^(S) s'annule au moins ä 1'ordre £+1 en λ.
DEFINITION 6. On appelle spectre d'une fonction f de ,tf (C) moyenne-periodique generalisee, note Sp(f), l'ensemble des couples (λ,£), λ G C, l e IN, tels que les fonctions Φ, . appartiennent ä V (f) pour 0 < j 0 £ et non pour j = £ + 1.
REMARQUES: Du theoreme de Hahn-Banach on deduit que le spectre Sp(f) est forme par les zeros communs aux transformees de Fourier-Borel generalisees des fonetionnelles analytiques de J^'(C) orthogonales ä V (f), chaque zero etant compte avec son ordre de multiplicite.
La proposition suivante que 1 ' on deduit du Theoreme 3 etablit un lien entre les fonctions moyenne-periodiques generalisees et les fonctions moyenne-periodiques classiques:
PROPOSITION 5. Une fonction f de Jtf(C) est moyenne-periodique generalisee si et seulement si la fonction X 1(f) est moyenne-periodique classique. Exemple: On a:
Χ - 1 ( Φ λ j^H2) = z e xP λ z.
Les fonctions moyenne-periodiques classiques ont et! etudiees par L. Schwartz [7] qui a introduit la notion de spectre d'une fonction moyenne-periodique f de J^(C): c'est l'ensemble des couples (λ,£), λ G C, i e IN, tels que les exponentiel1 es monomes z exp λζ appartiennent a V (f) pour 0 £ j ^ £ et non pour j = i + 1.
D'apres le theoreme de Hahn-Banach c'est aussi l'ensemble des
394 Κ. Τ. Meche zeros communs aux transformers de Fourier-Borel des fonction-nelles analytiques de J^'(C) orthogonales ä V (f), chaque zero etant compte avec son ordre de multiplicite.
Ensuite, il a demontre le theoreme qui caracterise les fonctions moyenne-periodiques classiques (voir [7] p 926, voir aussi [4]).
De ce theoreme et de la proposition 1. on deduit un theoreme de representation des fonctions moyenne-periodiques generalisees. Plus precisement on a:
Theoreme 4. generalisee JTC) par d
:
es
Soit f une f< Alors f peu t comb inai sons
Dnction de .# (O etre approchee 1ineaires
moyenne-period dans
f inies des la topolog fonctions
ique ie Φ.
de \ \
0 £ j _< Z9 telles que les couples (λ,£) appar t iennent a Sp(f).
2) Le groupe ^ .
Notation: On designe par:
-$„ (resp ^nni) le groupe des au t omorph ismes Y de J^(C) tels que: YL = LY (resp YD™ = DmY)
1/ (f) (resp M m(f)) le sous espace ferme de Ji?(C) engendre par Yf quand Y parcourt ^.(resp ^ n m ) .
Proposition 6. ([2] p 694, [6] p 227) : Tout element Y de a Dm
est de la forme:
Yf(z) = <S ,f> z
ou S est la fonetionnelle analytique de ^ f ( C ) donnee par:
J"0(Sz) (λ) = ZmrJ0 Μ*(λ) exp(X ü)jz) , ω = exp (liH)
avec a(j) a(j) Mj (λ) = m"m-1 + · · ' + - V " + M j ( X ) ' j = °» 1-"» m-1
Λ verifiant les conditions suivantes:
Fonctions Moyenne-perio di Ques Associees 395
m-1 , . >. Σ a U ; ü)PJ = 0 , 0 < p < s < m-1
j - o s "
Les fonctions M., j = 0,1,...,m-1, appartiennent a Exp(C) .
Le determinant de la matrice:
Μ*(λ) Μ*(λ) M m - 1 ( X )
M* (λω) Μ*(λω) M* 9(λω) m—1 u m- Δ
M*(Aü)m~1) M*(Xu)m""1) M*(Au)m~1)
est une constante non nulle.
Remarque:
Quand m = 1 on a:
Μ*(λ) = c exp(Xb) , b, c G C, c * 0
et par suite:
Yf(z) - c f(z+b)
PROPOSITION 7. : II y a identite entre les fonctions de J*f(C) gui verifient la condition:
i (f) * ,#xo D m
et les fonctions moyenne-periodiques classigues.
Demonstration : Le resultat s'obtient par une demonstration analogue a celle de la Proposition 3.1 de [2].
PROPOSITION 8.
1 ) Le groupe y est isomorphe au groupe ^
2) On a la relation:
396 Κ. Τ. Meche
t_(f) = X t (I-^f)) , f e Jt?(C). L Dm
THEOREME 5.: II y a identite entre les fonctions de ,Jt?(C) qui verifient la condition:
tL(f) * .JT(C)
et les fonctions moyenne-periodiques generalisees .
DEMONSTRATION
Le resultat decoule des Propositions 5, 7 et 8.
Ill. Developpement en Serie des Fonctions Moyenne-periodiques Generalisees par Rapport a une Fonctionnelle Analytique.
DEFINITION 6 . : Soit S une fonctionnelle analytique de JJ?f(C). On dit qu'une fonction f de ,>f(C) est moyenne-per iodique generalisee par rapport a S, si eile verifie l'equation de convolut ion :
S // f(z) = 0 , pour tout z e C
Nota tion: 1) on pose:
Z(J"(S)) = UAk,Jtk) , k G IN, itk G IN
οϋ λ, est un zero d'ordre JL +1 de la fonction entiere ^( S) .
2) On designe par Ψ11 Operateur de ,Xf' (C) dans lui meme, defini par : (III. 1) < ^/(T) ,g> = <T, r/(g)> , g G JT(C)
ou '/ = Χ ϋ Χ _ 1
II est cla ir que:
&(t(/\T)) (λ) = λί(Τ) (λ) .
Pour determiner des systemes biorthogonaux relatifs ä une
Functions Moyenne-perio di Ques Associees 397 f one t ionnel le analytique S de ,ϊ6% (C), on adapte a notre cas les methodes utilisees par L. Schwartz dans son etude des developpe-ments en serie des fonctions moyenne-periodiques sur IR (voir [7] p 882-889, voir aussi [9] 2 e m e partie).
1) Les zeros λ, , k G IN, de la fonction entiere J^(S) sont simples
PROPOSITION 9.: On considere les fonetionnelles analytiques t. , k £ IN, de ,tf (C) dont les transformers de Fourier-Borel generalisees sont donnees par:
1 . y(S)(X)
^(tk)U) = (JT(s)) '(Xk) " T ^
1
, si λ * λ.
, si λ = λ.
οϋ (JT( S) ) ' est la derivee de la fonction ^(S) . Alors ces fonetionnelles verifient:
1 si k = j i) <t. , Φ, n> = ό\ . , 6. . = k ' λ.,0 k , j ' k,j „ · , ^ · j ' J 0 J_L k * j
i i ) P o u r t o u t e f o n c t i o n f de , # ( 0 ) on a :
< t v , f > = k ' 1 ^ = ( ^ ( s ) ) ' ( x ; ) Τ7ΓΥ < s > χ ι . α " 1 ( ί ) ) >
ou I , , e s t l ' o p e r a t e u r d e f i n i p a r :
I . ( f ) ( z ) = / f ( t ) e k o
X k ( z - t ) d t , f G ,Xf(C)
DEMONSTRATION
Le resultat s'obtient par une demonstration analogue ä celle de la proposition (2.1.1) de [9], tout en utilisant l'operateur ^defini par la relation (III.1).
De cette proposition on deduit le resultat suivant:
PROPOSITION 10.: Si la serie
3 9 8 Κ. Τ. Meche
Σ C, η Φ, η (Ak,0)GZ (^(S)) k , U A k ' ü
converge vers f dans J^(C). Alors les coefficients C n , k € IN, sont donnes par la relation:
ck,o = <Vf>-2) Les zeros λ, , k € IN, de la fonction entiere ffi(S) sont d'ordre
Ak±i·
Notation: Soit F une fonction meromorphe ayant γ pour pole. On note F(A) "la partie singuliere" de F au voisinage de γ, c'est a dire la fraction rationnelle ayant pour unique pole γ, nulle a l'infini ei au voisinage de γ. nulle a l'infini et teile que F(X) - F ( X ) soit holomorphe
LEMME 1. On considere les fonetionnelles analytiques t. , k € IN de J<f'(C) dont les transformees de Fourier-Borel generalisees sont donnees par:
J"(tk)(X) ^(S)(X) · mshjrh> ι ί λ * h
, si λ = λ1
Alors, pour toute fonction f de >J^(C), et pour tout k € IN on a: L +1 k
<tk, f> = <qk #s, x ^ T 7 7 r o n ? k ( - 1 ( f ) )>
ou q, est la fonetionne1le analytique de ,lf(C) dont la trans-formee de Fourier-Borel generalisee est:
y<qk)(X) - U " X k ) V 1 j 7 T i J T I T
DEMONSTRATION
Une demonstration analogue ä celle du lemme (2.1.4) de [9] et 1 f u t il isa t ion de l'operateur ^donnent le resultat.
Fonctions Moyenne-perio di Ques Associees R e m a r q u e : S i l e s ze*ros λ. ,^- ,„ s o n t s i m p l e s , a l o r s on a : - k AGIN v 9
399
qk = (g(s))'(Xk)öo
οϋ όπ est la masse de Dirac au point zero.
et par suite: H - ck
PROPOSITION 11.: Parmi les fonetionnelles analytiques t, , k, m
0 1 m i K de jr'(C) verifiant:
k ,nr AV, j k,m k ,m
1 s i v = k , j = m
0 ailleurs
on a celles qui sont donnees par:
k ,m m - ^ <c* - v " h * k ,m # S
ou r est la fonetionnelle analytique de J^!(C) dont la trans-formee de Fourier-Borel generalisee est
(λ-λ ) m
^(Tk,m)(X) " W<S)<X)X (X-Ak)" f 1 m! V(S)(X)X,
DEMONSTRATION.
Le resultat decoule du Lemme 1. et d'une demonstration analogue ä celle de la proposition (2.1.3) de [9].
REMARQUE. Si les ze"ros X , k G IN, sont simples, alors:
r k, m
et par suite:
Lk,0 c k - c k
PROPOSITION 12.: Si la serie:
400 Κ. Τ. M e c h e
Σ ( X k , £ k ) G Z ( ^ ( S ) ) C k , Ä $ X k , £ k
c o n v e r g e v e r s f d a n s ^ ( C ) , a l o r s l e s c o e f f i c i e n t s C, , k , m
0 < m _< Z , k e IN, sont donnes par la relation:
(III.2) C, m = -V <(*<? - λ, ) m t ,f>. k ,m m! k k
DEMONSTRATION.
Le resultat s'obtient par une demonstration analogue ä celle de la proposition (2.1.4) de [9].
Soit f une fonction de ,)f(C) moyenne-periodique generalisee par rapport ä la fonetionnel1e analytique S de Jtf'(C). On considere la serie:
Σ C k l Φ λ % (in.3) (Ak,£k)e z(^(s)) K,Jt Ak,,6k
ou les coefficients C , 0 <_ m <_ & , k e IN, sont donnes par la relation (III.2). On peut alors se demander:
- La serie (III.3) converge-t-elle dans Ji?(C) ? et par quel procede ?
- Quand eile converge, quel est le lien entre sa somme et la fonction f ?
DEFINITION 7♦: On dit que la serie (III.3) converge vers f dans .tf(C) par groupement des termes, si on peut grouper les elements de l'ensemble Z(^(S)) en des sous ensembles finis Z., j = 0,1,.., de teile maniere que la serie:
oo Σ ( c Φ ) j=0 ak,Jtk)e z. k,x- Ak'*k
converge vers f dans ,j^(C) .
THEOREME 6.: La serie (III.3) converge vers f dans ,>^(C) par groupement des termes.
Fonctions Moyenne-perio di Ques Associees 401
DEMONSTRATION
D ' a p r e s l e TheOreme 3 . on a :
fcX(S) * X ~ 1 ( f ) ( z ) = 0 , p o u r t o u t z e e
Done, la fonction X" λ(f) est moyenne-periodique classique par rapport a la fonetionnelle analytique X(S) de ^ T ( C ) .
On deduit alors du Theoreme 9 de [1] ρ 138 (voir aussi [3]) qu'on peut:
- grouper les elements de 1'ensemble
z(^0 o üx(s)) = z(*-(s))
en des sous ensembles finis Z., j = 0, 1, 2,...
- trouver une suite de nombres complexes d, p de teile maniere que la serie:
Ä Σ [ Σ d . z exp λ, z]
j-0 (Ak,£ k)e Z j *'*-
converge vers X _ 1 ( f) dans »>f(C).
Le Theoreme 1 et la relation (II.1) impliquent que la serie: 00
Σ [ Σ d k l Φλ Z ]
k k j
converge vers f dans Jf(C).
Par suite, de la Proposition 12. on deduit:
d, = C, , 0 < m < L· , k G IN. k ,m k ,m — — κ
Dfoü le resultat.
BIBLIOGRAPHIE
1. C. Berenstein and B.A. Taylor, A new look at interpolation theory for entire functions of one variable, Adv. Math. 33
MAA—AA
402 Κ. Τ. Meche
(1979), 109 - 143 . 2. J. Delsarte and J.L. Lions, Transmutations d'operateurs
differentiels dans le domaine complexe, Comment. Math. Helv.
32, (1957) , 113 -128. 3. A.O. Gelfond, Linear differential equations of infinite order
with constant coefficients and asymptotic periods of entire functions, Trudy. Mat. Inst. Steklov. 38 (1951) 42 - 67; English transl, Amer. Math. Soc. Transl. 84 (1953) 1 - 3 1 .
4. A.F. Leont'ev, Series of Dirichlet polynomials and their generalizations, Trudy. Mat. Inst. Steklov. 39 (1951)
5. B.M. Levitan, Generalized translation operators and some of their applications, (1964).
6. N.I. Nagnibida, Isomorphisms of analytic spaces that commute with differentiation, Math. Sbornik, 72 (114), (1967) 221 -231 .
7. L. Schwartz, Theorie generates des fonctions moyenne-period-iques, Ann. of Mat. 48, (1947), 857 - 929.
8. F. Treves, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, Science Publishers, Paris, (1966) .
9. K. Trimeche, Fonctions moyenne-periodiques associees ä un Operateur differentiel singulier sur (0,°o) et devel oppemen t en serie de Fourier generalisee, a paraitre dans J. Math.
Pures et Appt.
ON AN APPLICATION OF REGULAR VARIATION IN PROBABILITY THEORY
J. L. GELUK Department of Mathematics, Erasmus University,
Rotterdam, Netherlands
Let / : H -* ]R be measurable. Then the function / is said to be
regularly varying with exponent a (notation: f€RV ) if f(tx) a
l im j>/+\ = % for a l l x > 0. £-><*> ?t)
Examples: t , t Lnt9 ta Lrft t (in In t) ( 3 , Y € 1 R ) are regularly varying with
exponents a.
Regularly varying functions play a role in Tauberian theorems concerning the Laplace transform.
A famous theorem of Karamata (see, e.g. [5]) states that if / is non-decreasing, then f€RV if and only if f (\ ft) £RV , where / is the Laplace-Stieltjes transform of the function /, defined by
oo
/(e) = J e~St df(t) (s > 0). (1) 0
A similar theorem can be proved for the transform / defined by oo
f(s) = in s j exs-fW dx (S> 0). (2)
0 If / is non-decreasing, then f£RVa with a > 1 if and only if /€i?7R» where a and ß are related by a~ + 3~ = 1 .
In order to formulate our results we need the so-called complementary function f ,
Suppose /: K. - K. satisfies f(x)/x-*<» for x-><». Then we define the function f* by
f*(s) = sup ys-f(y) for s>0· y> 0
403
404 J. L. Geluk Any f€RV has a finite complementary function for a > 1 and for a= 1 only if f(x)/x~*ao.
ry jL· a / ( a 1 ^
For example if f(x)=x with α > 1 , then / (s) = (a—l)(s/a) and (f*)*(x) = f(x) · This example suggests the extension f € RV with α>1 if and only if / €/?7R with β>1 where a and 3 are related by a-1 + 3_1 = 1 and (/ ) ~ f in this case.
This extension is correct under the assumption f is non-decreasing. See for example [2],[l].
The above results suggest a similar behaviour for the functions f and f* in case f G RVa with a> 1 . Indeed, we have f ~ f* for f G RVa (a > 1 ) as was observed by Bingham and Teugels [2]. In case a= 1 simpliar results can be obtained if the class RV is replaced by the class HRV (see definition 1 below). This is shown in [6], We can now state our first result: THEOREM 1. Suppose the random variable Y has an entire characteristic function. Suppose a, 3 > 1 are related as above. Then the following statements are equivalent: (a) P( \ γ \ > y) > 0 for all y G TR and there exists a function φGRVa such that
^ - * n P ( | j | > y ) = K (4) y-*oo φ(2/)
(b) There exists a non-decreasing function ψ€#7.,β such that
— (EY2n)l/2n ψ(2η) 1/3 -1/α ,_, lim - * — — = 3 e (5) n + oo 2^
In the above case f(v) ~<P*(y) (j/-»-) (6)
where ψ is an inverse function to ψ, defined for instance by \^x) = inf y;i>(y) > x\ .
Example. If Y is Ν(\ι,θ2) distributed, then a=3 = 2, $(y) ~ y2/2o2 (y ->oo) and ψ(η) - (2η)ΐ/2/σ (η->«>).
In order to prove Theorem 1 we take fx) =~ &n ( 1 -F(x) ) »where F is the distribution function of the random variable |Y|. Then f(s)=log- 1 + F(-s) by partial integration. Since the characteristic function of Y is entire we have
oo °° F(-s) = f eSX dF(x) = ^ ^ sn where μ = E\Y\H .
0 n=0 We use a result of Levin (Lemma 1) connecting the behaviour of the
coefficients of the power series of an entire function g with the behaviour of the maximum modulus. For the function q we take the characteristic
Application of Regular Variation in Probability Theory 405
function of the random variable \γ\ . Then we combine Levin's result with a result like that of Bingham-
Teugels mentioned above, in order to connect the asymptotic behaviour of the sequence of moments E\Y\ with the asymptotic behaviour of the tails of the distribution function of the random variable Y. In order to do that we have to replace the asymptotic equality / ~f above by inequalities (see Lemma 2).
LEMMA 1. (Levin [10], Ch. 1.13). Suppose CO
g(z) = £ cnzn
n = 0 is entire with maximum modulus
M(s ,g) = max \g[z) | (7) \z\ =s
and suppose ψ€2?7 ,R (B> l) is non-decreasing (then the inverse function
Under the above conditions
Tim- \cn\l/n^(n) = (£β)1/ 3 (8) W-»co
if and only if IS to^g»g> = l . (9) S->oo ψ (s)
LEMMA 2. Suppose F is the distribution function of a non-negative random variable with entire characteristic function. The function F is defined as in (1) above. Let φ : ]R -> H + be such that φ is finite. Then
ί(-8)<βΦ ( δ ) for s>0 implies 1 -F(x) < e~^ ( x ) for x>0. (10)
Moreover if φ € RVa with a > 1 , then, for any ε > 0 there exists s such that
\-F(x) <β~Φ(χ),χ >x0 implies F(-s) < e l + ε) Φ ( s ) for s > s£ . (11)
PROOF. For x > 0 , s > 0 we have
~F(x)<e'SX \ eStdF(t)<e-SXF(~s) < eHs)~
The implication (10) follows by taking the infinum over s > 0 on the right-hand side. In order to prove (11) we write
F-s) = \ eSXdF(x) + [\-FxQ)]eSX° + s \ 1 -F(u) eSU du . 0
By the hypothesis on F v.re have 0 x
0
406 J. L. Geluk
°° °° * S i-F(u)e8u du<s \ e
s"-*( M )du = e ( 1 + 0 ( l ) ) * (s)(S-~)
(see [2], section 4). Since 4>*€i?7g with 3 > 1 we have eCS = o(e^ (s)) (s-*») for e € l .
This completes the proof of (11). We use the next Lemma which is a consequence of Lemma 2 in order to
prove Theorem 1.
LEMMA 3. Suppose F and F are as in Lemma 2 and φ £ RV with a > 1 . Then
lim -Ml-ffr» = , (,2)
if and only if üii»it*i=I. (13) s_>oo Φ (S)
PROOF. Suppose (12) holds true. Then for any ε > 0 there exists aj£ such that -in(\-F(x)) > (\-ε) φχ) for a? > x£ .
Using (11) we get for s > s£
in F(-s) < (ΐ+ε)(1-ε) φ* (β/(1-ε)) .
Since φ^βθ-ε)"1) ~ (1—ε)"° φ*(β) (s-*«>) and ε > 0 is arbitrary, the above inequality implies
If
Tj- £nF(-s) lim < 1 . S-*oo φ (s)
I S £"f(-s) < i S-*<x> φ*(β)
we use (10) and φ** ~ φ to get a contradiction. The rest of the proof is similar. Proof of Theorem 1. Denote the distribution function of the random variable X = |Υ| by F. By Theorem 7.1.2 in Lukacs [10] we have M(s 9 g) = F (s)
where g is the characteristic function of X and the functions M and F
are defined as in (7) and (1) respectively. Application of the Lemmas 3 and 1 shows that (4) is equivalent to (8)
with ψ+ ~ φ* where ψ € Α 7 1 / β and \c |=#|γ|η/η!. Since (nl)1'*1 ~ n/e
(n-*oo) we find i s ( ^ D ' ^ w ^ ' / B . - ' / q . (14)
n
The equivalence of (5) and (14) follows since ( E \ Y \ ) is a non-decreasing function of n and ψ(2η) ~ ψ ( 2 η + ΐ ) (η-+οο). This finishes the proof.
Application of Regular Variation in Probability Theory 407
REMARKS.
(1) Davies [4] connects the behaviour of s log F(-s) (s -*<*>) with the behaviour of -y ΙηΡ\γ\ > y) (y -*<») in our notation. (2) The implication (5) ->(4) remains correct if we replace lim and lim in the theorem by limits.
PROOF. First we show that if we replace lim by lim in (8) and (9), then (8) implies (9). In order to prove this we may assume that for any ε > 0 there exists η(ε) such that for all n > η(ε) we have
1«„1>0-ε)η ^ V · Combining with Cauchy's inequality gives
MsW ,g) > \on\ sn)n > exp (n/3) > exp [ (1-2ε) 3 ι|Λ (s(n)) ]
for all n sufficiently large if we choose
Q-i/S , ( , a
s(w) = - ^ ^ (then n ~ ( 1 - ε) * M*~(s(n) )) for n=l,2,... 1 — ε
Suppose Sp (k = 1 ,2, . . .) is an arbitrary sequence tending to infinity as /c-»°°. Then for all k sufficiently large there exists n-, such that s(nA < s-, < s(n« + 1 ). Hence
lnMsk,g) ^ inMs(nk),g) lim > lim k->°° ψ (s.) k->co ψ (s(n^+l))
lnM(s(nv) , g) R = lim — 7 " > (ΐ-2ε)3 ,
/c-*oo ψ (s(«fe)) the equality ψ (s(n, + \)) ~ ψ (β(ηθ) (&-*«>) being justified by the uniform convergence theorem for regularly varying functions.
Since ε > 0 is arbitrary we find
> 1 lim in M(a,g) S ~*°° I**" ( )
Combination with (9) then gives lim £nM(s^)/f(s) = 1 .
s -*°° Now the proof of the statement can be completed as in the proof of Theorem 1 by noting that Lemma 3 with lim and lim in (12) and (13) replaced by limits coincides with Kohlbeckerfs result mentioned in the introduction. (3) The implication (9) -»(8) with lim's replaced by limits is incorrect as
2 the example g(s) = exp z shows. In Theorem 1 the characteristic function of the random variable Y is
408 J. L. Geluk an entire function of order 3 with 1 < ß < o°. It is well-known that the order of an entire characteristic function which does not reduce to a constant is at least equal to 1 (see Lukacs [ll] , thm 7.1.3). This leads to the question whether it is possible to formulate extensions of the above theorem to the cases 3=1 and |3=«>.
In order to formulate an extension for the case 3 = °° we have to consider a subclass of RV, defined as follows.
DEFINITION 1. We say φ€Π7?71 if φ(£)/£-«> (£->») and if there exists a positive function a such that
lim Φ<**>-*Φ<*> =χ ίηχ f o r a l l x > 0 . (15) £-♦<» ta(t)
EXAMPLE. φ(£) = t int, a(t) = 1 . For properties of this class see [7]. In order to formulate our second
result and the lemmas leading to its proof it is useful to define another class of functions which is a subclass of RVQ.
DEFINITION 2. A non-decreasing function ψ: H + -»]R+ belongs to the class Π if ψ(°°) = °° and if there exists a function a such that
ψ(£χ)-ψ(£) ,,,. lim τχτ = in x for all x > 0 . (16) , ^ a(t)
Notation: ψ€Π. The function a (determined up to asymptotic equivalence) is called an auxiliary function for ψ. It is well-known that (16) implies a(tx) ~ a(t) (£-»») for x > 0 and
t ta(t) ~ J sd\p(s) (£-♦») .
1 Moreover a(t) =0(ψ(£)) (t -«>). See [8], thm. 1.4.1.
From the above definitions it is easy to see that ψ€Π implies £ψ(£) € JIRV1 and conversely £ψ(£) £ JlRVl implies ψ £ IT if ψ is non-decreasing.
THEOREM 2. Suppose the random variable Y has an entire characteristic function. Then the following statements are equivalent: (a) P(\Y\ > y)> 0 for y £ It and there exists φ £ JIRV1 and a as in definition 1 such that
-inP\Y\>y)-$y) lim 7-r = 0 ·
(b) There exist ψ €Π and a as in defintion 2 such that
Application of Regular Variation in Probability Theory 409
r (g|nW)-l/w(*l)1/w-»(n) „ lim = 0 .
TT^O 01(77) In this case the functions φ and ψ are related by
s φ*(β) -(es)'1 J \^u)du (s-°o).
0 EXAMPLE. Y is Poisson distributed with parameter λ . Then
φ(χ) = xinx—x — x in λ + ο(χ) (#-<») , a(x) = 1 , ψ(η) = in n-£n £n n - in X- 1 + o( 1), a(w) = 1 .
The proof of Theorem 2 above is similar to the proof of Theorem 1. We only formulate the analogues of the Lemmas 1, 2 and 3 and omit the proofs. The most difficult part is the analogue of Lemma 1 above. In order to formulate this analogue we need one more class of functions. DEFINITION 3. A non-decreasing function γ: ]R -> ]R belongs to the class Γ if there exists a function β : TR -* E. such that for all x € ]R
Y(*+*g(t)) Ä lim — = e . (17) t->oo Ύ(*)
EXAMPLE. ψ(α) =£ηαχ + of^" 1^) (a > 0) and γ(χ) = (1 + o( 1)) eX / a (a > 0). It is well known that ψ€Π if and only if ψ €Γ . See [8], thm. 1.5.5.
It is possible to replace 3 in (17) by a function which is asymptotically equal. Moreover the function 3 satisfies 3(t) = o(t) (t -*«>) . See [8], Lemma 1.5.1. In case Υ = ψ the auxiliary functions a (in (16)) and 3 in (17) are related by
s ^ / ψ (u) du
α(ι|Λ(β)) - 3(e) ~ - (β—»). Ψ (s)
For further properties of the classes IT and Γ the reader is referred to [8]. LEMMA 1'. Suppose the function
/ (« )= y czn n
ft=0 is entire with maximum modulus
M(s9f) = max \f(ss)\ . \z\=6
The following statements are equivalent: (i) There exist ψ and a satisfying definition 2 such that
Ι*ηΓ1/Μ-Ψ(") 1™ α(») "°· (,8)
410 J. L. Geluk (ii) There exists a function γ satisfying definition 3 such that
inM(s,f) l i m γ(β) = '· ( 1 9 )
S -><» In the above case the functions ψ and γ are related by
s γ(β) ~ (es)"1 J ^ u)du (8-+co). (20)
0 LEMMA 2'. Statement (11) in Lemma 2 is correct under the assumption φ€ ITi?7,.
L£MM4 3'. Suppose c|):JR+-*]R , <\>t) ft is non-decreasing, φ € TIi?F1 satisfies (15) and F and F are as in Lemma 2. Then
lim = 0 (21)
if and only if
Πϊ M£l2*l = 1 . (22) S->oo φ*(δ)
REFERENCES
1. A.A. Balkema, J.L. Geluk and L. de Haan, An extension of Karamata's Tauberian theorem and its connection witn complementary convex functions Quart. J. Math. Oxford 30 (2) (1979), 385-416.
2. N.H. Bingham and J.L. Teugels, Duality for regularly varying functions, Quart. J. Math. Oxford 26 (3) (1975), 333-353.
3. N.G. de Bruijn, Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform, Nieuw Arch. Wisk. 7 (1959), 20-26.
4. L. Davies, Tail probabilities for positive random variables with entire characteristic functions of very regular growth, Gamm Tagung Göttingen 56 (1975), 334-336.
5. W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, John Wiley, New York, 1966.
6. J.L. Geluk, L. de Haan and U. Studtmiiller, A Tauberian theorem of exponential type, Canad. J. Math. (1986), (to appear).
7. J.L. Geluk, Τΐ-regular variation, Proc. Amer. Math. Soc. 82 (1981) 565-571.
8. L. de Haan, On Regular Variation and its Application to the Weak Convergence of Sample Extremes, Math. Centre Tract, 32, Mathematisch Centrum, Amsterdam, 1970.
9. E.E. Kohlbecker, Weak asymptotic properties of partitions, Trans. Amer. Math. Soc. 88 (1958), 346-365.
10. B.J. Levin, Distribution of zeros of entire functions, Amer. Math. Soc. Translations, 1964.
11. E. Lukacs, Characteristic Functions^ Griffin, London, 1970.
HILBERT SPACE STRUCTURE ON THE SOLUTION SPACE OF HELMHOLTZ
EQUATION
H. L. MANOCHA Department of Mathematics, IIT, Hauz Khas,
New Delhi — 110016, India
1. Introduction. This paper, based on the contributions made by the author in [2,3,4,5], is devoted to obtaining a new biorthogonal basis in terms of the J-function for an infinite-dimensional solution space of the two-variable Helmholtz equation (3 +3 +ω )Ψ0Ε,£/)=Ο, ω > 0 . We introduce at the out-
xx yy
set in the next section, Section 2, the J-function and construct a dynamical symmetry algebra for it from the differential recurrence relations obeyed by it. Thereafter, we make use of this symmetry algebra to form a canonical system of partial differential equations having J-function as their common solution. To be precise we show that the J-function is a solution of the two-variable Helmholtz equation as well as of the three variable heat equation.
In Section 3, we study the imbeddings of the J-function into the two variable Helmholtz equation and characterize it as eigen function of a mixed first and second order operator Q in enveloping algebra of the conformal symmetry algebra E(2) f the Lie algebra of the Euclidean group E(2).
In Section 4, we deal with a solution space F of the Helmholtz equation which is an isomorphic image of LAS.) , the Hubert space of Lebesgue-square integrable functions defined on the circle 51 , under the Fourier transformation. The Hilbert space structure on LAS.) induces the Hilbert space structure on F. With the help of the operator Q which is not self-adjoint, we derive biorthogonal basis for the subspace, consisting of all twice differen-tiable functions, of L20S^), anc* then correspondingly for the space F, indeed in terms of the J-operator.
411
412 H. L. Manocha 2. Canonical Form. We introduce the function
J f, i rn m
^ ^ a > w + n ^ ) w mini m ,n = o
( a ) = a(a+ 1) · · · (a + n - 1 ) when ηΦθ , (a) Q = 1 .
( 2 . 1 )
L e t
J = J[cL9$;x,y] w^w2 . (2.2)
Following the method described in [2 ,5 ] , we define the d i f f e ren t i a l opera
tors
i y
£ = Μ 1 « 2 3 Λ . ,
( 2 . 3 )
a l a i *
w„ 3 2 U ,
The action of these operators on J i s
^a3 I 1 J f a+1
J ( a + i , 3+1 ) J =
K a
_ x ß J J =
' f a
3
( a - 1 ) J(a-\ )
( B - i ) J ( B - l )
J .
Further, the equations (2.4a) yield the i d e n t i t i e s
(2.4a).
(2.4b)
where
ClJ = °2J = °'
0 = Ε α Ε - \ , C = ECL-E0ECL[1
l a ' 2 β (2.5)
for general values of the parameters a and 3· It may be noted that the E-
operators are independent of those parameters. The notation E , for example, is merely used to indicate that the action of the operator on a basis function J is to raise the α-parameter by 1 and multiply it by Γ* However, the operator makes sense when applied to any differentiable function of the four
Hubert Space Structure 413 variables x9y ,u, ,u2 , Now a basis function J for fixed values of a , 3 i s characterized by the equations (2.Ab) and (2 .5) . Indeed, the equation (2.4b) determines the dependence of J on the variables u,9u„:
J = Jx%y)u[u\ . (2 .6 )
Substituting this expression in (2.5) and factoring out the u-dependent terms, we obtain the system of differential equations for J~[a,3; x,y]:
^♦»WV'-0· (2-7) (*3«B + ß 3 * - V J = 0 ·
To within a constant term, the only solution of equation (2.7) analytic in a commutator neighbourhood of (x , y) = (0,0) is J[a , 3; # , £/] ·
It may be seen that the E- and Z-operators close under the commutator bracket to form a 6-dimensional Lie algebra G and that the J are simultaneous eigen functions of the commuting i£-operators. Further, the elements of G are symmetries of the system of partial differential equations C .J= 0,
j = 1,2, that is, if φ is a solution of the system and L£G , then £φ is a solution of the system.
To cast the system into a canonical form we employ Lie theory. It may be noticed that the 4 -operators in (2.3) form a basis for an
abelian subalgebra of G and that this subalgebra is locally transitive in its action on the minifold with coordinates (χ,ζ/,ζ^ , u2 ) with the exception of a singular submanifold of dimension <4. Furthermore, by means of the multiplier transformation
K-^K = pK , E-*E = pE p"1 ,
where the operator p corresponds to multiplication by u~ u~ , we can eliminate the wT1 term in the expression for E and u~l term in the expression for E^ so that the operators E become purely differential. It follows immediately that there exists a coordinate system (V tv ,V ,V ) such that
F1 = 3j , Ea = 32 , Ea(i = 33 , 5β = 94 , 3.. = 3υ ; (2.8) Equations (2.5) take the canonical form
(a12-i) J = (3i-33J J = o . J = u~l u0
l J , (2.9)
where x = V2 V3 Vh ' y = Vl V2 » Ul = V2 ♦ U2 = Vk ' ^2, 10^
In other words
414 H. L. Manocha m — 1 R — 1
J =J[a,&;v2v3v^,vlv2] v 2 vh
is a simultaneous eigen function of the canonical equations '12 1-0 , 31-331 +~ 0
as well as of the eigenvalue equations
D2-D1-D3~a-\ , Dk-D3~ 6-1 ,
D. = V.d . , £ = 1,2,3,4.
(2.11)
(2.12)
(2.13)
3. Imbedding of the J-Function. Here we study the imbeddings of the e7-function that arise from the fact that one of the canonical equations takes the form
(312-ι)Ψ(υ1,ν2) = 0. (3.1) Introducing new variables
ζλ = τνλ + τν2 , ζ2 = νλ-ν2 , ( 3 . 2 )
(3.1) takes the standard form
(\l2l+\z2
+ l ) ^ z l ' z l ) = ° · (3.3)
The symmetry algebra of (3.3) is the Lie algebra G generated by P , P , M]
where [6] Pl = h ^ P2 = \ ' M = 2 2 3
3 l - 2 1 32 2
satisfying the commutation relations
[P^Pj-0, [M,Pl]=P2, [M.pJ.-Pj. (3.3) may be rewritten as
(ρ^ + ρ2 + ι ) ψ ( 2 ι 5 3 2 ) = o
Treating z and 22 as real variables it may be noted that G is an isomor-phic image of £(2), the Lie algebra of the Euclidean group E(2), having, [6],
(3.4)
(3.5)
(3.6)
0 0 0 0 0 0 1 0 0
P/ -
0 0 0' 0 0 0 0 1 0
, M'
0 - 1 0 1 0 0 0 0 0
(3.7)
as its basis, where
[P;,P2] = 0 , [M',P'1] = P'2 , [M',P2,]=-P1
/ . (3.8)
Furthermore, the representation T of E(2) on the solution space of (3.3) is given by, [6],
[T(g) ψ] ζλ 9z2) = ψ(3χ cos d + z2 sin0 + a,
-ζγ sin6 + 2 cos Θ + b ) , (3.9)
H u b e r t S p a c e S t r u c t u r e
cos Θ - s i n 6 0
s i n Θ cos Θ 0
a b 1 £ E(2)
415
(3.10)
a,b € l ä 0 < θ < 2π . For the imbedding of the J-function, we note from the eigen value equation (2.13) that
D3^D2-D1-a+\ , Ζ^-Ζ^-Ζ^ + β-α (3.11)
when acting on a basis function; hence we replace 33 and 8^ in (2.12) by expressions on the right hand sides of equations (3.11) and set, for convenience, v λ —v. = 1. We conclude that the functions
1 i 2 J[a,$ ; -\izY+ iz2) , ~\ (ζ2λ+ζΙ)] (izl + z, xa-i (3.12)
satisfy the Helmholtz equation (3.3), and are characterized by the eigenvalue equation
(ίρλ + P2 — a — -Ltd) ( a — 1 + iM) •0. (3.13)
Thus the solution (3.12) is characterized by a pair of operator equations in terms of first and second order operators in the enveloping algebra of E(2) .
If we put 2, = r sin Θ , z^= r cos θ , we find that
F = J[a + 1 ,3; - \ rt9 ~ ^ r 2 ] r a t a , t =-
satisfies the equations
(P+P~-l) F = 0 ,
[ρ++(3-α-1-Ρ°)(α + Ρ°)]Ρ = 0, where
P"
ίΡχ-Ρ2 = t(dr-r~ltdt) ,
^ - p 2 = t
iM = £9,
-1 -3 -r~l £9.) , r t' *
(3.14)
(3.15)
(3.16)
4. A New Basis for Functions on the Plane. We rewrite the Helmholtz equation (3.3) as
where (tfj /) are Cartesian coordinates. The Lie derivatives
P = 8 2 2/ M a x y
(4.1)
(4.2) 1 X
form a basis of the symmetry algebra of (4.1). The equation (4.1) has been studied in [6], Chapter 1, from the view
point of separation of variables.
416 H. L. Manocha Let Sl be the unit circle e%$ : — π < φ < π , mod27r and L2(*S'1) be the
Hubert space of Lebesgue square integrable functions h on S, with respect to the measure ά,φ, We extend any 7z € L (£ ) to the real line by the periodicity requirement h($+2T\) = /ζ(φ) . The inner product on L (S ) is defined as
π hi>hi)=\ Μ φ ) Μ Φ Μ Φ , hjeL
2(si) ' (4*3) - π
Then, for each h €L (S ) , the function π
Ψ(#>#) ~ IW = exp [ίω(χ cos Φ + z/ s i n φ) ] /ζ(φ) 6?φ (4 .4 ) -π
is a solution of (4.1). The Euclidean group E(2) acts on the solution ]\)(x,y) via the opera
tors T(g) as defined in (3.9): π
T[g)Hx9y) = \ exp[<iu(xcos<$> + ysin<$))][T(g)h]<\))d<l· ( 4 .5 ) - 7 T
where
[T[g)h]M = exP [>ω (acos (φ-θ) + bsin (φ-θ))] Τζ(φ-Θ) . (4.6)
Thus the operators Tg) induce operators, which we also call Tg) , acting on h and given by (4.3).
A simple computation yields
Hg)hi .Hg)h2) =h1>h
2) < 4 · 7 > for all h.EL (S ) , that is, the operators T(g) are unitary.
The Lie derivatives (4.2) induce, via (4.4), corresponding operators on L2(5:) :
P, = ίω cos φ , P0 = £ω sin φ , Af = — . (4.8) 1 z <2φ
Operators (4.8) define an irreducible unitary representation (ω) of E(2) on i2(5x) [7,8].
We now restrict ourselves to the space F consisting of all solutions ψ of the equation (4.1) such that ψ(#,ζ/) = 1(h) for some h € L (S ) . Here F
is a Hubert space with inner product
(Ψ1.Ψ2) = (Äj.Äz) . Ψ - = Ahj) (4.9)
Note that each function \\)(x,y) in F can be considered as inner product ψ(α?,2/) = 1(h) = (h,H(x,y ,. )) ,
#(#,ζ/,φ) = exp [— ϊω (x cos φ + y sin φ)] € L 2(S i) · (4.10)
It is well known that the Helmholtz equation separates in four coordinate systems, the corresponding separated solutions characterized as eigen
Hubert Space Structure 417 functions of the seGond order operators in the enveloping algebra of E(2)
In [6], Chapter 1, the spectral resolution of these operators have been computed in the L2(51) model and the mapping J, (4.4), has been used to derive expansion for the separated solutions of the Helmholtz equation. Here we list the (obvious) resolutions for the two most elementary of the four systems, those corresponding to polar and Cartesian coordinates:
2 (i) Polar M . A basis for L2(S1) consisting of eigen functions of M is
m ίηφ
n'w = V - >n= > ±] > ± 2 · · · ( 4 · η > Mf(l) = - i n f ( 1 ) , ( f ( l ) , / ( ΐ ) ) = δ „ ,
Ψ ^ ν , θ ) = ΐ ( / η( 1 ) ) = ΐ η / Γ ι ^ Η e x p ( i n e ) ,
x = r cos Θ , y = r sin Θ . 2
( i i ) Cartestan P .
f f (Φ) = δ ( φ - λ ) , - π < λ < π , (4.12)
ο Λ2) - Λ A2) /A2) Λ2)\ χΐ\ \ / \
ΨΛ \ χ >y) = I(f ) = exP [ί ω( ^ c o s λ Λ-y sin λ) ] . Α λ
Here J (r) is a Bessel function. Spectral resolutions for the parabolic cylinder and elliptic systems, as well as overlaps relating these systems, can be found in [6].
It follows from (3.13) that the solutions of (4.1) which are also eigen functions of the operator
Q = M2 + ξΜ+ω"1 [ίΡγ +Ρ2) ,
ξ a complex constant, can be expressed in terms of «/-functions defined in (2.1). (We assume in the sequel that iE, is not an integer.)
Defining Q on the subspace of twice continuously differentiable functions h in L (S ) , by (4.13) and (4.8), we find a basis for LAS ) consisting of eigen functions of Q :
/(3)(φ) = *ξ φ / 2.Γ, .Ale-^'2), n = 0,±l, ±2, ·.. (4.14)
Qf3) =nil-n) f(3) . ^J n v - ' J n
For a proof that the eigen functions of Q form a basis even though Q is not self-adjoint, see [l], Chapter 12.
418 H. L. Manocha
The adjoint operator Q* also has a basis <7^ of eigen-vectors:
*ί3)(Φ) = < Γ ζ Φ / 2 ' Λ2* i<$>/2\ m = 0 , ±1 , ± 2 , -(2τη + ϊξ)
Q*g3) = m(il-m)g(3) .
Clearly these two bases are biorthogonal. Computing (f^3' , g^ M to find
the normalization, we find
(4.15)
<4 3 ) >4 3 ) ) = 2 sin ίηζ
ίξ-2η nm
The corresponding solutions of the Helmholtz equation are /0 in(π/2-θ) . , 0.w
^ , θ ) = J(/<3), = ί2Ε£ <ω*/1>_
(4.16)
(4.17) Γ(2η-£ξ+1) Γ(η+1)
n+1, 2η—£ξ + 1 ;
•2πg '(ωρ/2
io)2? -i6 a) r 2 e 4
(4.18) Γ(-2η-τ:ξ+1) Γ(-ΤΗ+1)
-m+\ , -2ιη-ϊξ+\ ;
where ρ,θ are polar coordinates. These expressions make sense for all integer values of m, n since
Γ(α)_1 J"[a,3; #,£/] is an entire function of a. This leads us to the following theorem:
THEOREM. Each fEF can be expanded uniquely in the form oo
, ( 3 ) f = y b F ( 3 ) ,
where
ίζ-2η b = n 2 sin inE, (f*G. ( 3 ) >
(4 .19)
(4.20)
(· > · ) is the inner product on F. The series (4.19) converges in the Hubert space sense.
REFERENCES
1. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
2. E.G. Kalnins, H.L. Manocha and W. Miller, The Lie theory of two variable hypergeometric functions, Studies in App. Math. 62 (1980), 143-173.
3. E.G. Kalnins, H.L. Manocha and W. Miller, Transformations and reduction formulas for two-variable hypergeometric functions on the sphere £2 » Studies in Appl. Math. 63 (1980), 155-167.
4. E.G. Kalnins, H.L. Manocha and W. Miller, Harmonic analysis and expansion formulas for two-variable hypergeometric functions, Studies in App. Math. 66 (1982), 69-89.
Hubert Space Structure 419 5. H.L. Manocha, Lie algebraic characteristics of two-variable hypergeome-
tric functions, Froc. of the Int. Symposium on Algebra and its Applications, Marcel Dekker, Vol.91, 1984.
6. W. Miller, Symmetry and Separation of Variables, Addison Wesley, Reading, Mass., 1977.
7. G. Mackey, Induced Representations of Group and Quantum Mechanics, W.A. Benjamin, New York, 1968.
8. N. Velinkin, Special Functions and Theory of Group Representations (translation from the Russian) Amer. Math. Soc. Trans. (Vol. 22), Amer. Math. Soc, Providence, R.I. 1968.
TEACHING THE ART OF APPLYING MATHEMATICS
R. R. McLONE Faculty of Mathematical Studies, University of Southampton,
Southampton, UK
The commonest way of educating, particularly at university level (and its equivalent in polytechnics, institutes of technology and similar institutions) is to follow tradition. In Britain, as in many countries, the majority of those graduating have passed through a similar system — lectures, tutorials, and timed examinations. The philosophy for this approach appears to be to ensure that the "best" students are brought to the frontiers of knowledge. But the question repeatedly raised in recent years is "what about the rest?", a question which has increased relevance when one considers that it is only a very small proportion of the graduates in a modern university who are in any way concerned with these frontiers. So, as a preliminary to the main part of this talk, I would like us to consider some of the principles on which one might build an undergraduate mathermatics curriculum appropriate for a modern university.
First, then, what might be considered to be the aims of a 'Mathematical Education'? Following Griffiths [4], one might expect that it would be to get students
— to pass examinations in mathematics — to perform mathematical tasks — to be a full human being.
(None of these need be mutually exclusive, although the greatest pressure in many education systems is to lay stress on the first.) However whatever the response it will be shaped by the relationship of the responder to the society in which he(she) is placed — one cannot talk about even the design of a mathematical education in a cultural way, as though it were divorced from the nature of the community in which it takes place. Why should we expect
Invited talk.
421
422 R. R. McLone society to support our desire to provide a mathematical education of the type generally provided? Why does society believe that mathematics 'is important1
at all levels of education? Pollak in his introduction to "Teaching and Applying Mathematical Model
ling" (Berry et dl, [2]) has no doubt as to the reason: "Not because mathematics is beautiful — which it is — or because it provides great training for the mind, which many mathematicians believe it does; but because it is so useful". Now if usefulness is indeed to be a fundamental reason for teaching and studying mathematics, then we as teachers are obliged to show, exercise and emphasize that usefulness at every opportunity. Moreover, what is taught and how it is taught must change as the needs of society change. The advent of the almost universal availability of the microcomputer and the calculator will also profoundly affect the mathematics curriculum, if not mathematics itself. For example there will be an emphasis on such topics as exploratory data analysis and the theory of estimation; and an emergence of the algorithmic point of view as an organising principle. But in addition this usefulness and the added impact of new technology serve to empahsize that mathematics is an activity more than a body of knowledge and it is this concept that lies at the root of "the art of applying mathematics".
But what is "mathematics"? Who is "a mathematician"? The answer to the first of these questions, generally involves asking the second, and vice Versa. Generally, mathematics is defined by its content — analysis, calculus, algebra, probability, ... and so on. Even this definition becomes blurred at the edges — to what extent is statistics a part of mathematics? Even more controversially, what about such subjects as operational research or computing science? Anyway we cannot put every bit of known mathematical content in an undergraduate course for no other reason than student stamina is not up to it! Is mathematics better defined as a battery of techniques? Or perhaps its essence lies in theorem and proof, setting up axioms and the conduct of rigorous argument. Or does it lie in its capacity to be servant to so many other disciplines? There is no single answer — it depends on who is responding to the question, and in what context.
As I discovered as long ago as 1973 in an investigation into the educational qualities and knowledge required by mathematicians ("The Training of Mathematics"), employers, students, graduates, academic mathematicians all take differing views. However, apart from the last named group (the academics) there was at least a clear consensus that the undergraduate mathematics curriculum at the time was failing the student, the graduate, and his subsequent employer. What was significant however was the agreement between all these
Teaching the Art of Applying Mathematics 423 groups that the precise content of the undergraduate course was not as important as the mathematical skills and abilities required to operate as mathematicians in practice. Many of these abilities were simply developed in an under-graduate course.
Yet, at least as I consider what goes on at my own university on a normal day in term, the predominant concern in an undergraduate course is the . mathematical content. Indeed such syllabuses as are set down, for example in university prospectuses, refer almost exclusively to particular pieces of mathematics —matrices, determinants, eigenvalues, differential equations, and so on. Now a good lecturer can bring these to life, but in so doing almost certainly calls on something more than a mere recitation of theory. Indeed content lists can induce an attitude in the lecturer that what matters is that the syllabus is 'covered1; and an expectation in the student that what matters is to pass the examination at the end of the course.
But I reiterate mathematics is something which is practised, not a dead body of theory; and that practice is in the context of the real world in which we all live. Moreover the changes that occur in the way mathematics is practised, and the nature of the mathematics used are not just whims of fashion, but are responses to social, political and technical circumstances. Until the Second World War mathematics graduates largely remained in academic life (or taught in schools); over the last 40 years they have first moved into government establishments (often as a consequence of the war effort — e.g. the origins of operational research), then into industry — starting with aircraft and associated industries, but becoming involved in research and development in other industries also — and, over the past 10-15 years especially, into commerce, finance, etc. The nature of the work of mathematicians in their employment has also changed. They take on non-mathematical activities; often being required to show how problems can be amenable to mathematical treatment which otherwise might appear not so, and especially to talk to and be involved with the practical side of their organisation (engineers, production managers, accountants, etc.)
Above all the mathematics used must be fitted to the real world and not the other way round.
When considering what practising mathematicians require when pursuing their craft, what do we find? Industrialists usually reply with general statements such as
(i) literacy (ii) numeracy (iii) ability to communicate with an emphasis on conciseness
424 R. R. McLone (iv) ability to work in a team (v) accepting responsibility (confidence in one's ability).
The listing of such attributes is no less applicable to mathematicians for the fact that a similar list could well apply to any professional group working in an industrial context. A list of attributes more directly related to mathematicians, at least in a British context, is seen in "The Training of Mathematicians". [One must say again that the social and cultural context does matter, and influences the result; however an understanding of what skills are required in work can be achieved whatever the context (cf. Griffiths [4]).] The following 'areas of ability' were identified as being important:
Tat
Problem formulation Problem solution Communication — oral Communication — written Critical evaluation
)Le 1
Acquisition of new knowledge Wide range of knowledge Overall planning Ingenuity Accuracy
and techniques
Balance* + 2 + 58 -20 -71 - 10 + 63 + 27 -42 + 23 + 30
^Difference between those reporting the ability 'present' and those reporting it 'markedly absent1 in mathematics graduates.
Certain features of Table 1 are worthy of note. There is the clear inability to communicate, whether orally or in writing, which often leads employers to comment on the difficulty of introducing mathematics graduates into management; an observation underlined by the often reported difficulty in getting students to talk about their subject, even to other mathematics students. The Table also indicates an inability to plan a programme of work. Mathematics graduates entering employment are accustomed to seeing their work in terms of the immediate task at hand (almost as though it were a tutorial problem) , and find it difficult to see it in the overall scheme of things. Above all, although the graduates could solve posed problems, the ability to formulate them was as much absent as it was present, and this in an educational tradition in which 'Applied Mathematics' features as strongly as 'Pure Mathematics'. In some respects, however, this is due to a view of applied mathematics which sees it as a study of particular theory and the solution of standard types of problem. [A description of the employer's view of the
Teaching the Art of Applying Mathematics 425 average mathematics graduate might be summarised: Good at solving problems, not so good at formulating them, the graduate has a reasonable knowledge of both literature and technique; he has some ingenuity, and is capable of seeking out further knowledge. On the other hand he is not particularly good at planning his work, nor making a critical evaluation of it when completed — and in any event he has to keep it to himself as he has little idea how to communicate it to others.] How can the 'balances1 in Table 1 be adjusted so that future graduates are better prepared to be 'useful1 mathematicians — useful that is to society and worthy of their hire? Can we turn these 'areas of ability1 into something which can be related to the undergraduate curriculum? (And still, of course, brings the 'best' to the frontiers of knowledge).
A traditional syllabus, as we have already noted, consists of a sequence of contents which are to be covered during the course. This emphasis on content stems from a desire to ensure that the student acquires enough material to understand subsequent courses. However we are now looking for some way of identifying these skills which are pertinent to the way a mathematician works. Research at Southampton, and elsewhere, identifies four general areas of ability exhibited through mathematics: Technical^ Discovery_, Critical and Communicative. Let us consider these abilities.
Technical skills — for example, the acquisition of basic mathematical techniques, and theory; the use of these techniques to solve standard problems — are naturally found in all degree courses. On the other hand the skill of using existing techniques in unfamiliar situations is less evidently developed although it also might be described as essentially one of technique. However, the other general abilities we have identified find little place in a standard undergraduate course. The Discovery skills — improvisation of new techniques when existing ones are inadequate, formulation of problems (recall Table 1), even the abstraction of a unifying principle or the simplification of a complex situation both of which are often lecturer-led, with little practice for the student — are found in few standard courses. Discovery skills are, however, at the heart of mathematical activity, and especially in the application of mathematics. True, they do need 'time', but so also do our students need the experience. Good discovery methods develop imagination and initiative in students whose performance in standard lecture courses appears weak; and for whom new insights into mathematics rekindles enjoyment and motivation. [Indeed, and somewhat ironically, these abilities are developed with great teaching skill in elementary school mathematics, before the content dominated curriculum takes over.]
Other skills equally important but generally left out of the under-
426 R. R. McLone graduate programme are those of Criticism and Communication. Students find the development of critical skills difficult, except at a superficial level. Although the organisation of material from books, papers, etc. is standard in other disciplines, e.g. the humanities, geography, even the natural sciences, mathematics students are usually excused the activity, it generally being done for them by the lecturer. Assessing the suitability of different mathematical models, asking pertinent questions in a mathematical situation of their own and others' work are not easy skills to assimilate. Moreover they are especially difficult if associated with mathematical technique and theory which is at the limit of a student's competence, as is usually the case on the few occasions when required. Communication skills have clearly in the past been found wanting; even when needed in a direct mathematical form. Indeed if the only practice is in terms of tutorial answer sheets and examination papers, there is not much surprise in the graduates1 lack of expertise. Communicating results to non-mathematicians and working in a group, with each member playing a complimentary and effective role, are not part of the standard curriculum.
Now the art of applying mathematics contributes significantly to the acquisition of mathematical skills which are often more important than knowledge of the content of any particular course. Let me attempt to reformulate the above skills into specific attributes which could be regarded as a starting point for setting out some goals in teaching the art of applying mathematics. The acquisition of these attributes (first expressed in a conference -report on university mathematics teaching (UMTC. [9]) would go a long way in-preparing a mathematics student for future practice. 1. Ability to identify situations and to formulate problems which are
suitable for mathematical treatment. 2. Ability to handle and make sense of natural or experimental data.
Representation and interpretation of data. 3. Determination of variables and parameters with which to describe obser
vation. Understanding the nature of these variables, e.g. random or deterministic, logical or numerical, discrete or continuous. Ability to select significant variables.
4. Translation of information (e.g. data, verbal description) into pictorial form (e.g. graph, diagram, flow chart).
5. Recognition of patterns in data and in processes. 6. Generation of mathematical expressions to summarise observations
whether directly or through the paths suggested in 4 snd 5. 7. Ability to set up a model representing the system and relating its
Teaching the Art of Applying Mathematics 427 significant variables. Use of previous experience of a variety of models to select good models.
8. Technical ability to manipulate the mathematical expressions of the model to achieve desired objectives.
9. Ability to consult books, journals, etc. for additional mathematical techniques or extra non-mathematical information as required.
10. Understanding of when to change a model, method or objective in discussing a problem. Recognition that numerical or computing techniques may be essential at some stage.
11. Recognition of what constitutes a solution and how this depends on the problem being considered and the objectives. Evaluation of success of models (e.g. significance of differences between solution and observation) and decision about further refinements.
12. Ability to work effectively in a group. 13. Ability to communicate clearly, especially in writing. Ability to
cooperate with non-mathematicians at all stages of model building including interpretation of final results. Let us consider the first of these. We tend to define all of the prob
lem boundaries in traditional applied mathematics and make the necessary simplifying assumptions before it is presented to the student. This is usually done to make the problem mathematically tractable and accessible to the student. Unfortunately this removes from the problem any chance of developing the ability to identify what the situation is that might be amenable to mathematical treatment, or to formulate a mathematical statement or model from it. Exercises which involve talking round a problem (maybe an ill-posed or vague one), refining it, and requiring the making of decisions on what to look at and what to ignore or neglect — without necessarily proceeding to a solution — can be important starting points for this ability.
Turning to another attribute (number 11), it is a terrible constraint on finding realistic or interesting problems that they be solvable. It is important for the student to have to face up to situations where they have to decide whether they give rise to 'solvable' problems — even what a 'solution' is. How can simplifying assumptions be justified other than on the pragmatic grounds that the problem cannot be solved otherwise? Students should be encouraged to keep returning to the original problem setting to compare the predictions of their model with what actually occurs. (Can one structure examples in this way within traditional courses? It is possible (see D'inverno and McLonet5]), but it needs careful thought, and then is only partially successful. It is so easy to predetermine a single, acceptable
428 R. R. McLone
solut ion.) What can be done positively to develop these attributes in our students?
Since such a wide range of skills is involved, it is clear that the majority of students, especially in the early years of their courses, will be unable to cope if they are faced with situations demanding a large number of them at once. Indeed experience suggests that a large number of students find great difficulty in coping with (1), without involving any other. Initially presenting mathematical situations involving just one or two, but progressing to work which involves a facility with many in one large problem is a possible way forward for any particular course. For instance, some strategies might be: (i) Presenting illustrations and examples of formulated models — for moti
vation purposes (this last phrase is fundamental — illustrations which do not illustrate do no one any good).
(ii) Introducing necessary mathematics to give confidence in tackling a given problem (i.e. relating the mathematics to the problem as a needed tool, rather than in isolation; this can be a useful technique in 'pure mathematics', too).
(iii) 'Mathematising' extra-mathematical situations lying within the direct experience of the students — for first year students this can mean quite elementary mathematics (from school) — to give mathematical confidence.
(iv) Confronting students with problems illustrating the inadequacy of their 'problem solving' schemes (e.g. which surface is rougher — that of the earth, or an orange?).
(v) Confronting students with problems showing inadequacy of their 'already learned' mathematics (with some hope for motivation of new mathematical material).
(vi) Insisting that students communicate, defend, and validate their own models and solutions.
(vii) Using small group work with criticism and communication first within the group (a step to achieving (vi), perhaps).
(viii) Introducing various kinds of project-oriented courses, possibly including industrial/commercial (or any practical) experience — or a detailed case study.
(ix) Involving cooperative lectures from specialists in the substantive field of a project or projects.
(x) Involving visits from industrial or other 'outside' speakers to demonstrate the activity of applying mathematics in a practical environment.
Teaching the Art of Applying Mathematics 429 Whilst in the first year of the course one might concentrate on applications which demonstrate the skills (1) —(13), e.g. is a (i) , in subsequent years one seeks to encourage students to tackle more and more work which follows the strategies just set out. [As a 'mathematical modeller' I believe that the activity of modelling achieves just this.] (See for example, Andrews and McLone [l], Berry et al [2], Burkhardt [3], James and McDonald [6], McLone [8]).
I would like to conclude this talk by remarking that, as many of our graduates will eventually teach at some level, it seems reasonable that one of our aims should be to induce an awareness in them of the teaching problems they will meet: one such is allied strongly to the idea that it is skills/abilities which are central to teaching and occurs whatever the level of teaching. It is what Griffiths [4] has called the "Problem of the Three Languages". As teachers we are often required — and especially at university level — to teach a piece of mathematics which is initially and formally expressed in some "official" language of the Mathematical Community, such as has been presented at this Conference, or what constitutes published research papers in learned Journals. Let us call this F . This 'language' includes not only the vocabulary and grammar of mathematics, but also other qualities which are acquired by the participants (i.e. lectures) — consistency, questioning, belief that it matters, etc. On the other hand the target audience has a language C — colloquial perhaps, but reflecting that portion of society from which the students (pupils) come. A teacher has to infer how rich or poor C may be by suitable probing; indeed his own natural language may be quite strange to C, with consequent room for misunderstanding. The teacher's job is to invent a 'bridge' language — τ — a sort of transform from one to the other. Diagrammatically we might write
T(T) C(S) v > F(M) .
The primary aim is that the student can cross the bridge from one language to another. In most examples of undergraduate teaching, the university academic is tempted to go straight to F, to hasten the transport of the 'best' to the frontiers of knowledge. Unfortunately many are lost in the process. Even within our traditional undergraduate teaching there is always a need to find the appropriate τ. The real challenge however is to find the transform which takes the analogue of C(5) which is the students' perceived skills and abilities and turns them into the analogue of F(M) — those attributes which make up a practising mathematician in the real world. Then the university teacher will be teaching the art of applying mathematics.
430 R. R. McLone
REFERENCES
1. J.G. Andrews and R.R. McLone, Mathematical Modelling, Butterworth, 1976. 2. J.S. Berry et al, Teaching and Applying Mathematical Modelling, Ellis
Horwood, 1984 (Proceedings of an International Conference on Mathematical Modelling held at Exeter, U.K. in 1983).
3. H. Burkhardt, Learning to use mathematics, Bull. Inst. Math. Appl. 10 (1979), 238-243.
4. H.B. Griffiths, Aims of a mathematical education, Proceedings of 2nd South East Asia Conf. on Math, Educ. , 1981, 19-26.
5. R.A. d'Inverno and R.R. McLone, A modelling approach to traditional applied mathematics, Math. Gaz. 61 (1977), 92-102.
6. D.J.G. James and J.J. McDonald, Case Studies in Mathematical Modelling, Stanley Thornes, 1981.
7. R.R. McLone, The Training of Mathematicians, Social Science Research Council, 1973.
8. R.R. McLone, Teaching mathematical modelling, Bull. Inst. Math. Appl. 10 (1979), 244-246.
9. UMTC (Undergraduate Mathematics Teaching Conference), Using and Learning, Shell Centre, Nottingham, 1979.
