ELSEYIER

APPLIED NUMERICAL

MATHEMATICS Applied Numerical Mathematics 24 ( 1997) 13 1-148

An extrapolation method for a Volterra integral equation with weakly singular kernel

Pedro Lima, Teresa Diogo * Departamento de Matemdtica, Institute Superior Tknico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal

Abstract

In this work we consider second kind Volterra integral equations with weakly singular kernels. By introducing some appropriate function spaces we prove the existence of an asymptotic error expansion for Euler’s method. This result allows the use of certain extrapolation procedures which is illustrated by means of some numerical examples. o 1997 Elsevier Science B.V.

Keywords: Volterra integral equations; Euler’s method; Asymptotic expansions; Richardson’s extrapolation

1. Introduction

We consider the second kind Volterra integral equations

t

u(t) = I K(t, s)p(t, s)u(s) ds + f(t), t E (0, T], 0

p(t,s) := J- 1 SW 1

J;;Jp@pp’ and

t

u(t) = I K(t, s)q(t, s)u(s) ds + g(t), t E (0: T]>

* Corresponding author. E-mail: teresa.diogo@math.ist.utl.pt.

0168-9274/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PIISO168-9274(97)00016-O

(1.1)

(1.2)

(1.3)

(1.4)

132 I! Lima, T. Diogo /Applied Numerical Mathematics 24 (1997) 131-148

where p > 0, K(t, s) is a smooth function and f and g are given functions. Equations of this type arise from certain heat conduction problems (cf. [ 1,121). The special case of (l.l), with K(t, s) = 1, was studied theoretically in [12] where a somewhat complicated Lz solution was derived, by using a method involving Watson transforms. In [8,1 l] some analytic expressions valid in weighted &-spaces were obtained, but none is useful if a tabulated solution is required. Further results were obtained in [ 131, where it was shown that a unique solution to (1 .l) exists in Cm[O, T] if K(t, s) = 1 and p > 1. In [6] the general equations (1.1) and (1.3) were considered, with

fqt, t) = 1, K(t, s) = constant, K@, s) = /L(t).

There some results on existence and uniqueness of solution in the spaces C”[O, T] were presented and some regularity estimates were derived. In particular, when K(t, s) = 1 and 0 < CL 6 1, Eqs. (1.1) and (1.3) were shown to have a family of solutions in C[O, T] of which only one has C’ continuity.

We note that the functions p(t, s) and ~(t, s) in (1.1) and (1.3) are integrable with respect to s in [0, T]. However, unlike most weakly singular kernels, they do not satisfy t t J p(t, s) ds + 0 and J q(t, s) ds + 0 as t + O+.

0 0

This fact creates some difficulties in the convergence analysis of numerical methods for the above equations. Two low order product integration methods have been studied in [ 131, where approximations to the solution of (1.1) of orders one and two were obtained with the, so called Euler and trapezoidal methods, respectively. In [5] a fourth order method was developed which employs cubic splines in the continuity class C’ [0, T]. Eq. (l.l), with K(t, s) = 1, was shown to be equivalent to (1.3), whose kernel &t, s), although unbounded, is more tractable than p(t, s). The approximate solution was then required to satisfy both Eq. (1.3) and its differentiated form at the mesh points (Hem&e-type collocation). High order product integration methods have also been investigated in [4]. Although the convergence results in the above mentioned papers were obtained for the case K(t, s) = 1, they can be extended to the general equations (1.1) and (1.3).

In the present work we shall be concerned with the numerical solution of (1.3) by a low order method in conjunction with extrapolation procedures. Extrapolation methods based on the midpoint rule have been studied in [7] for nonlinear Volterra equations with smooth kernels. In [lo] a comprehensive analysis of extrapolation methods and their application to various types of functional equations is presented. In particular, an extrapolation method based on the trapezoidal rule is studied in detail.

For the sake of simplicity, we shall assume that K(t, s) = 1 in (1.3). However, it is easily seen that the techniques used in this paper can be extended to the general equation. This work will be organised as follows. In Section 2, we transform Eq. (1.3) into a new equation, so that the two equations are equivalent away from the origin. Moreover, under appropriate conditions, although for a certain g the solution of the original equation may be unbounded at the origin, the corresponding solution of the new equation will be smooth. In Section 3, Euler’s method is used to approximate the solution of the new equation. Following the approach developed in [lo], we show that an asymptotic error expansion is valid which permits the use of certain extrapolation procedures. This allows us to improve the low accuracy of the approximations obtained by Euler’s method. The theoretical results are confirmed by some numerical experiments.

P: Lima, T Diogo /Applied Numerical Mathematics 24 (1997) 131-148 133

2. Existence and uniqueness of solution

Let us consider the following equation:

Nt) = g(t), t E (0, Tl,

where

h(t) := u(t) - j- su(s) ds.

0

Associated with (2.1) is the following equation:

t

v(t) = ./

&--I--P Tv(4ds +W,

0

where p 3 0 is a real number and

v(t) := t&(t), g(t) := @g(t).

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

Introducing

t

_Lpv(t) := v(t) - s $4--1--P ---v(s) ds, ill-P

0

we can rewrite (2.3) as

Lpv(t) = g(t).

We start by introducing some definitions.

(2.6)

(2.7)

Definition 2.1. For T > 0 and m a non-negative integer, Vm[O, T] denotes the normed space of the real valued functions f such that f E Cm[O, T] with

(2.8)

Definition 2.2. For T > 0, ,B 3 0 and m 3 0, V,,p[O, T] denotes the normed space of real valued functions f such that tPf E Vm[O, T] with

(2.9)

Definition 2.3. Let m 3 1. Then f E Vi[O, T] if and only if f E Vm[O, T] and f(0) = f’(0) = . . . = f’“-“(O) = 0. I f m = 0, we have V$[O,T] = Vo[O,T].

Definition 2.4. Let m 3 1 and p 3 0. Then f E Vz,,[O, T] if and only if tPf E Vz[O, T]. If m = 0,

we have V&[O, T] = Vo,p[O, T].

134 f? Lima, Z Diogo /Applied Numerical Mathematics 24 (1997) 131-148

We see that Vz[O, T] c Vm[O, T] and that Vz p[O, T] c V,,p[O, T], for every ,8 > 0, m 3 1.

Moreover, it is easily proved that Vi[O, T] c V$[O, T], for every ,D > 0, and that V~,,[O, T] = Vz[O, T], for every m 3 1. On the other hand, for every m > 1 and p > 0, there are elements of V,,p[O, T] that do not belong to Vm[O, T].

Throughout this paper, m will always be a non-negative integer and T > 0, /3 3 0 will be real numbers. The following theorem has been proved in [13].

Theorem 2.1. Zfp > 1 in (2.1) and thefunction g in (2.1) belongs to Vm[O, T] then (2.1) has a unique solution u E Vm [0, T].

The following theorem extends the results of Theorem 2.1 to the spaces Vm,p[O, T].

Theorem 2.2. Let p > 0 and p > 1 + ,8. (i) If thefinction g in (2.1) belongs to Vm,p[O, T] then (2.1) has a unique solution u E V,,,[O, T].

(ii) If g E Vz,,[O, T] then (2.1) has a unique solution u E Vz,,[O, T].

Proof. (i) We follow the proof technique used in [ 131. For any u E Vm,p [0, T], we define w = SPu such that

t

w(t) = I SF- 1 tll 4s) ds + g(t)-

0

By a change of variables s = ta, we obtain

1

w(t) = I up-‘u(ta) da + g(t).

0

Multiplying both sides of (2.11) by tP yields

@w(t) = s

(t~)h&+r~-~-~ da + #g(t).

0

Thus, for 0 < j < m, we have

$ (@w(t)) = i 2 [(t&(ta)] crp--1--p da + -$ (@g(t)), Vt E [O,T].

0

Setting s = ta, we may write

(2.10)

(2.11)

(2.12)

(2.13)

2 [(tu)bL(ta)] = $ [d%(s)] 9. (2.14)

NOW let ul,u2 E Vm,p[O,T] and WI = Sp(ul), w2 = Sp(212). Then, from (2.14), (2.13) and an application of the mean value theorem for integrals, we obtain

R Lima, ir: Diogo /Applied Numerical Mathematics 24 (1997) 131-148 135

From (2.15) it follows that

IlWl - w211/3,m < +1 - ~2llP,rn~

(2.15)

(2.16)

Since p - ,0 > 1, (2.16) implies that S, is a contraction mapping on the Banach space V,,p[O, T]. Therefore S, has a unique fixed point u E V,,p[O, T], which is the solution of (2.1) in this space.

(ii) Let us now prove the existence of a unique solution of (2.1) in Vz,p[O, T]. Since this is a subspace of V,,p [0, T], we just have to show that

SPu. E Vz,,[O, T] whenever u E Vz,,[O, T]. (2.17)

In order to prove (2.17), we let t ---f 0 + in (2.13) and, using the mean value theorem for integrals and the fact that tpg E Vi [0, T], we get

1

tliF+ $ (tpSpu(t)) = IliF+ -$ [(t<)pu(t[)] 1 dpl-p+j da, j =O,l,..., m, (2.18) 4 i

0

’ where < E [0, 11. Since ‘u. E V, p [0, T] the limit on the right-hand side of (2.18) exists for j < m,

being equal to 0 when j < m - ‘1. Hence SPu E Vz p[O, T] and this concludes the proof. 0

Remark. An alternative way of proving part (i) of Theorem 2.2 could be the following. If g is a function of V,,p [0, T] then 9, defined by (2.5), belongs to V, [0, T]. Now, since p > 1 +p, Theorem 2.1 can be applied to (2.7) and we conclude that this equation has a unique solution u E V,[O, T]. But there exists only one ‘u. E V,,,[O, T] such that u(t) := v(t)/t p and, of course, u is the solution of (2.1).

Example 2.1. Let us consider Eq. (2.1) with 1 -cost

SW = fi .

Since t’/2g(t) = 1 - cost E V:[O,T], then g E Vl,,, [0, T]. If p > 3/2, then it follows from

Theorem 2.2 that (2.1) has a unique solution u E I!:,,, , ‘[W’].

We now give two lemmas which will be used in the next section.

Lemma 2.1. rf g E Vz,,[O,T], with m 3 1, then g’ E Vz_,,p [0, T]. In general, if m 3 k then the

kth order derivative of g belongs to Vz_,,,[O, T].

’ Proof. Since g E Vm,p [0, T], we have that g(t) := tog E V$ [0, T]. Therefore, using Taylor’s theorem, we may write

@g(t) = sg’“‘(O) + r&t), b’t E [O,T], (2.19)

136 R Lima, 7: Diogo /Applied Numerical Mathematics 24 (1997) 131-148

where r, = o(P). Dividing both sides of (2.19) by tp, we obtain

g(t) = trn-P -(m) 7g (0) + t-%&), v’t E (O,T].

From (2.20) it follows that

g’(t) = (m - P)t”-P-’ ij’“‘(0) - pt-P-G,(t) + t-PT’ (t) m! m 7 vt E (0 T] ,

(2.20)

(2.21)

(2.22)

Finally, multiplying both sides of (2.21) by tp, we obtain

tPg’(t) = Cm - PP”-’ -(m) m!

g (0) - @-‘Tm(t) + r:,(t), vt E [O, T].

Since r,(t) = o(P), by applying the 1’Hospital’s rule we may conclude that

/3-%,(t) + r&(t) = O(P).

Therefore, from (2.22) it follows that @g’(t) E V$-1 [0, T]. Thus, g’ E Vz_I,a[O, T]. Using this result repeatedly, we may conclude that if g E Vz,PIO, T] and m 3 k then g@) belongs

to Yj&p [O, T]. ??

Lemma 2.2. Let g E V&[O, T] and let v be a real number: If k + [v] 3 0, then P’g E IJ’~+~~~,~[O, T], where [v] stands for the integral part of u.

Proof. We have, with g(t) := @g(t),

tk (k) @g(t) = g Yj (0) + o(t’E+l).

Multiplying both sides of (2.23) by t”, we obtain

tP+Yg(t) = tkfu -(k) 7 g (0) + 0(t”+“f’).

Differentiating T times both sides of (2.24), we may write

$(t”+“g(t)) = (k + v) . . . (k + Y - T + 1)7,(k)(O) + O(tk+v-r+l),

wherer- 1,2,..., k + [v]. From (2.25), it follows that tVg E V~+l,l,P[O,T]. 0

3. Euler’s method and asymptotic error expansion

3.1. Euler S method

(2.23)

(2.24)

(2.25)

In this section, we show that an approximation to the solution u of the original equation (2.1) can be obtained by applying Euler’s method to the transformed equation (2.7). Let us assume that the function g in (2.1) belongs to Vi,a[O, T] for some ,B 2 0 and m > 0, and let ,B > 1 + ,B. Then

the function ij in (2.7) is in V$[O, T]. From Theorem 2.2 it follows that (2.1) and (2.7) are uniquely

P Lima, T Diogo /Applied Numerical Mathematics 24 (1997) 131-148 137

solvable in the spaces VA BIO, T] and Vz[O, T], respectively. Moreover, the solutions u and u of these equations can be related through (2.4) for t > 0.

In order to approximate w on [0, T], let us define an uniform grid Xh, with stepsize h := T/N,

Xh := {ti = ih, 0 6 i 6 N}. (3.1)

Setting t = tk in (2.3) gives

tk

V(&) = tp J ~~-‘-~,(s)ds +?j(t/J, 1 < k < N. (3.2) 0

We first consider the case when u is bounded at t = 0. Then we approximate the integral on the right-hand side of (3.2) by a product integration rule

tk

J

k-l

s~-~-~,(s) ds P c Div(ti),

0 i=o

where

&+I

Di := J ,P-1-P ds=hp_p(i+ l)@-p - ill-p

P-P , l<k<N-1.

t,

(3.3)

(3.4)

The case when g and u are unbounded at the origin can be dealt with in a similar way, by noting that the limits

g(O) = iliy+ @g(t) +

and

v(0) = tliy+ t@,(t) +

(3.5)

(3.6)

are finite. So even if u becomes infinite as t -+ Of, we can still use (3.3). In order to obtain v(O), let t --+ O+ in (2.3):

v(O) - tliF+ tPep 4 J s~-'-~,(s) ds = s(O).

0

Since u E Vz p , [0, T] and p > 1 + p, it may be shown that

t

lim+ G-p J 0) s~-~-~zJ(s) ds = - P-P’

0

Substituting (3.8) into (3.7) yields

V(0) =?(o)P;--pP.

(3.7)

(3.8)

(3.9)

138 P Lima, 1: Diogo /Applied Numerical Mathematics 24 (1997) 131-148

From (3.2) and (3.3), we obtain the following equation:

k-l

?J; = g-p k c &$ + g(tk), 1 < /C < N,

i=o (3.10)

where V; := v(0) and V; is an approximate value of t{u(tk), 1 < k < N. Then we set

u; = z&f, l<k<N,

where u; is an approximation of U(tk).

(3.11)

3.2. Convergence and asymptotic error expansion

In this section, we shall prove that Euler’s method given by (3.10) is convergent when 9 E Vm[O, T]. Moreover, we shall show that, under appropriate conditions, the error allows a certain asymptotic expansion. We first introduce some notations. Consider the vector space

IL??+’ = {v” = (L&V:,. . . , v;)~, v” E IR, 0 < i 6 N},

with the maximum norm

(3.12)

Define the linear restriction operator rh : V,,p[O, T] + IEN+’ by

(r’f(t))i := f(ti), 0 < i < N,

and let L$ : RN+’ t IF?+’ be the discrete linear operator given by

(3.13)

(L/Lwh)k := k = 0,

1 < k 6 N, (3.14)

so that the approximate solution VJ~, defined by (3.10), satisfies

Liv” = rhg. (3.15)

Using the results contained in [lo], the method (3.15) is convergent and an asymptotic error expansion for the error 21h - rhn is valid if the following conditions are satisfied:

(1) (2)

(3)

For every function ij E V,“[O, T], (2.7) has a unique solution in Vj [0, T] , k = 0, 1, . . . , m. The method (3.15) is stable in the following sense: there exists a constant M, independent of h, such that for every Lj E Vz[O, 7’1, the approximate solution uh to the exact solution v of (2.7) satisfies

/uhlj G Mllrhgll. (3.16)

There exist functions C, E Vz_,[O, T], 1 < 4 < m - 1, such that

m-1

($(rhu)) - Th(Lpv))k = c C&k)@ + h(tk), 1 < k < N, (3.17) q=l

where r,(t) = O(hm).

P. Lima, T. Diogo /Applied Numerical Mathematics 24 (1997) 131-148 139

From Theorem 2.2 it follows that (2.7) satisfies the condition (1). We now give a sufficient condition for (3.16) to hold.

Theorem 3.1. If?j E Vo[O,T] and p > 1 + /3, then d satisfies (3.16).

Proof. Taking modulus in (3.10) gives

k-l

Since p > 1 + p, we may write

k-l cl-0

c Q = t,

i=o P--P’ l<k<N.

Substituting (3.19) into (3.1 S), yields

ll~hll G IVgll + Pll4L where p := 1 /(p - ,B) < 1. Thus

llvhll G &- IIrhd.

(3.18)

(3.19)

(3.20)

(3.21)

Hence, (3.16) is satisfied with A4 = l/(1 - p). •I

Before examining (3.17), we prove a convergence result.

Theorem 3.2. If 9 E Vo[O, T] and ,u > 1 + p then

IIqi(rhu) - m$y))) ---f 0, as /L-o+, (3.22)

lluh - AJII --+ 0, as h + o+. (3.23)

Proof. We first prove (3.22). From (2.7) and (3.15)

(L; (r’u) - rh(Lp)) k = tf-’ 5 ( :‘d-‘-b(s) ds - II&)) , l<k<N. (3.24) i=o t 2

Using the mean value theorem for integrals yields

&+I L+l

s s~-~-‘%(s) ds = w(&) J #-l-P ds 7 t, t,

with & E (ti. &+I), 0 6 i < k - 1. Substituting (3.25) into (3.24), we obtain

k-l

($ (~“u) - @(L/w)) k = SE-’ c Di (v&i) - v(Q), 1 < k < N, i=o

(3.25)

(3.26)

140 fl Lima, Z Diogo /Applied Numerical Mathematics 24 (1997) 131-148

which yields, after taking modulus,

(3.27)

Since 21 E Vo[O, T] then

o~~xN Iv(&) - v(ti)l + 0 as h --+ Of. \\

This gives (3.22), which means that the method (3.15) is consistent. In order to prove (3.23), we write

L; (d - Al) = L$Jh - Lj (AJ) = rhg - L$ (7%). (3.28)

Now an application of Theorem 3.1 yields

llvh - Thllll < lqr’l(Lpu) - L$(r”v) II (3.29)

and (3.23) follows from (3.29) and (3.22). In other words, we have proved that stability together with consistency imply convergence. 0

Corollary. Ifthefunction g in (2.1) belongs to $$[O, T] f or some p satisfying p > 1 +/I, then Euler S method by given by (3.15) is convergent.

Proof. We have that S = tpg belongs to Vt [0, T] and convergence follows from Theorem 3.2. 0

Theorem 3.3. Let 21 and z? be the solutions of(2.7) and (3.15), respectively, and let 1-1 > 1 + ,D, with fi > 0. If u belongs to Vz [0, T] f or some m 3 1, then the following asymptotic expansion is valid:

m-l

(_$(T”u) - T~(LBvJ))~ = c C&)hq + O(hm), 1 < k < N, (3.30) q=l

with C, E V$_,[O,T].

Proof. From (2.7) and (3.14) it follows that

(L; (r”u) - T’(L/~)) k = tf-” j? sp-‘-pz+) ds - 2 D&j), l<k<N. (3.31)

0 i=o

Using the variable substitution s = ti + ah, we may write

tk

J #-‘-b(s) ds = h E ](& + ah)P-‘-P,(ti + ah) da.

0 i=o 0

On the other hand, from (3.4), we obtain

(3.32)

(3.33)

P. Lima, T. Diego /Applied Numerical Mathematics 24 (1997) 131-148 141

Combining (3.32) and (3.33) gives

t!%

J sp-'%(s) ds - 5 D&) = h 5 1 (v(ti + ah) - v(ti)) (ti + ah)‘“-‘-

0 i=o i=o o

-0 da. (3.34)

Using a Taylor’s expansion, we may rewrite (3.34) in the form

tk

J

k-l

d-‘-%(S) ds - c Diw(ti)

0 i=o

= yf hr+’ ]Q&)$ dT+‘)(ti + ah)(ti + c~h)p-‘-~ da + 0(/P+‘), r=o 0 i=o

where

O,(u) := g! (-v+‘. In order to obtain (3.30), we shall need to analyse the sum

i=o

taking into account the smoothness of the functions involved. We start by defining

G(s) := ‘?,(t,@),

so that we may write

tk

J p-P ' tW-l--P&+l)(+jt = trfl J sW-l--P@(T+l)(S)& 0 Ic 0

We now consider two cases.

(3.35)

(3.36)

(3.37)

(3.38)

(3.39)

(i) We first assume that p - p is an integer and apply the Euler-MacLaurin summation formula to the integral on the right hand side of (3.39) (see, e.g., [3]). If hk := h/tl, = l/lc, then we have for each 0 E [0, I],

1

J k-l

Sp-‘--P@(r+‘)(S) ds = hk c ((i + fl)%k)P-l-B@(‘+l) ((i + o)hk)

0 i=o

m-r-2 -

c q=o s=l s=o

)

+o(hy-), o<r<m-1, (3.40)

where Bk(s), k = 1,2, . . . , are the Bernoulli polynomials. Then, expressing hl, in terms of h and using (3.38) yields

142 P Lima, I: Diogo /Applied Numerical Mathematics 24 (1997) 131-148

h k-l

-ID tf-” ti + oh)@-l-&(r+l) (ti + ah)t;+l

i=o

= t;+t m-T-2 hQ+’

scL-l-p@(T+l)(s) ds + tQ+’ c k

q=. (q + I)! Bq+l(g) 0

x ; (#-l--Py(T+l)(&.)) ) ( .S=l - ; (d-l-PdT+l)(Stk)) /

s=o )

+o((;)m-T), o<r<m-1.

Combining (3.41) and (3.39) gives

k-l

h c(ti + ah)~-l-b(r+l)(ti + ah) i=o

tk

= m-T-2 hQ+’ I?,+1 (CT) tp-l-@,(T+l)(t) & + t;-” c -

0 q=o t;+’ (q + l)!

;st(s"-1-31'(+1) s k _-

s=l (t))/ )

s=o

(3.41)

(3.42)

Substituting (3.42) into (3.35) and using (3.31), we obtain an expansion which is of the type of (3.30). To prove this, we derive an expression for the coefficients C,, 1 < q < m - 1. Let us define

1

CY T .- .- I 13,(a)da, 0 6 T 6 m.

0

Collecting the terms in h, we easily obtain that

tk

Cl (tk) = tf-‘aa I t’+‘w’(t) dt.

0

We may define

t

c,(t,) := to-‘,O .I s~-~-~w’(s) ds,

0

(3.43)

(3.44)

for t E [O,T]. Since u E V$[O,T], then Cr E Vi_,[O,T]. An expression for Cj(t), with j > 2, can be obtained in a similar way. Assuming that m 3 j we

define

P. Lima, 7: Diogo /Applied Numerical Mathematics 24 (1997) 131-148 143

(3.45)

with t E [0, T]. S‘ mce ‘u E Vz[O, T], then the first term of (3.45) belongs to L$&[O, T]. In order to prove that Cj E Vz_j [0, T], we need to analyse the following expression:

&T-2

dsi_T_2 (s~-~-‘&+‘) s ( t))( &r-2

s=l - F(S-?!(~+‘) s ( t))i

s=o (3.46)

for each T with 0 6 T 6 j - 2. j 3 2. Applying Lemma 2.1, we have that, for each t E [0, T],

F~-W+l) (st) E v;_,+,_p_2 [O, T]. (3.47)

Making again use of Lemma 2.1, we see that dj-V-2

dSi_T_2 (#-D-l&+1) (St)) E v;+,-,-j [O, Tl,

as a function of the variable s. From Definition 2.3 it follows that the second term of (3.46) vanishes. Then, using the product differentiation rule, (3.46) reduces to a sum of terms in the variable t, all being in the class V”

cj E V~_,[O,T].

m+k_P_j [0, 2’1. Finally, from (3.45) and an application of Lemma 2.2 yields that

(ii) We now consider the case when ,u-,8 is not an integer. By using the generalized Euler-Maclaurin summation formula for functions with algebraic singularities (see, e.g., [3,9]) an expansion for (3.37) can be obtained as follows. We have, with & := h/t,, and @ as in (3.38),

1

./

k-l sP-‘-~@p(T+‘)(s) ds = hl, c ((i + a)&)“-‘-‘&+‘) ((i + a)&)

0 i=O

7(-p + p + 1 - q, c+++‘+qo)

- (-L)qY(-q, 1 - ~)~(6p-1-~~(~+1)(8))1 z=l

) +0(“T--‘). (3.48)

Here ~(a, s) is the periodic generalized zeta function defined by

$a, s) := [(a, S), s - S = integer, 0 < 3 6 1,

where <(a, s) is the generalized Riemann zeta function given by

<(a, s, = go (s +&)

sin[27rslc - 7ru/2] ka+l , Reu>O.

144 R Lima, 7: Diogo /Applied Numerical Mathematics 24 (1997) 131-148

Now c$@+~) (0) = ~(~+l) (O)t;+l and, since u E Vz[O, T], then cD(~+‘+‘J)(O) = 0, 0 < 4 < m - 2. Finally, after expressing hk in terms of h we arrive at the following expansion:

k-l

C( ti + ,h)+‘%(‘+‘) (ti + oh) i=o

tk m-r-2 hq+l 1

= tp-l-pw(r+l)(t) dt + t;-" c tP+’ 3 y(-q, 1 - c); (s~-~-1&+1)(,tk)) 1

0 q=o /c s=l

(3.49)

After substituting (3.49) into (3.35), it can be shown that (3.31) possesses an expansion which is of the type of (3.30). The proof is similar to the one used for the case when CL - /? is an integer, therefore we omit it. 0

We are now in a position to state the following theorem which provides the basis for Richardson extrapolation.

Theorem 3.4. Let w and wh be the solutions of (2.7) and (3.19, respectively, and let p > 1 + ,f3, with ,B 3 0. Zf ‘u belongs to V$[O, T], f or some m 3 1, then the following asymptotic error expansion is valid:

m-l

(u” - ~~2,)~ = c C;(tk)hq + O(hm), 1 < k < N, (3.50) q=l

with Ci E Vz_,[O, T].

Proof. The above expansion follows from conditions (l)-(3), after an application of [ 10, Theorem 2.1, p. 181. 0

Remark. We note that Eqs. (2.1) and (2.7) are equivalent away from the origin. Therefore the con- vergence results for (2.7) are still valid for (2.1) in the above sense.

4. Numerical results

In this section the theoretical results of the previous sections are illustrated by means of some numerical examples. Following the remark of last section, we shall be interested here in verifying the convergence results for (2.7). The algorithm (3.10) was applied to the auxiliary equation (2.7), with

q(t) := tPg(t), g(t) := t” 1 - ( -$), ww> (4.1)

for several choices of the constants a, p and CL. The corresponding exact solutions of Eqs. (2.1) and (2.7) are u(t) = t” and w(t) = taf@, respectively. The norms of the errors in the approximate solution

P. Lima, Z Diogo /Applied Numerical Mathematics 24 (1997) 131-148 145

Table 1 Results for Examples 4.14.5, llell = lIrhu - ~1~11

Example 4. I Example 4.2 Example 4.3 Example 4.4 Example 4.5

h II4 II4 II4 II4 II4 h = l/40 O.l4D-0 O.l6D-1 0.22D- 1 O.l4D-1 0.27D- 1

h = l/80 O.lOD-0 0.82D-2 O.l2D-1 0.69D-2 O.l4D- I

h = l/160 0.73D- 1 0.41D-2 0.59D-2 0.35D-2 0.72D-2

h = l/320 0.52D- 1 0.21D-2 0.30D-2 O.l8D-2 0.36D-2

rate k N 0.49 k 2 0.98 k e 0.97 k CY 0.98 k CX 0.98

Table 2 Using the Richardson’s extrapolation to improve the numerical approximation of u(l), in Example 4.2

i S, = Td”’ T(“) I

T(“) 2

T(Z) 3

T(%) 4

0 O.l6D-1 0.23D-3 0.30D-5 O.l7D-6 0.20D-7

1 0.82D-2 0.60D-4 0.52D-6 0.29D-7 0.35D-8

2 0.41D-2 O.l5D-4 0.90D-7 0.5 lD-8

3 0.21D-2 0.39D-5 O.l6D-7

4 O.lOD-2 0.99D-6

5 0.52D-3

uh to ‘u are displayed in Table 1 for several values of the stepsize h. The following quantity has been used as an estimate of the convergence order:

(4.2)

In the cases when an asymptotic error expansion of the type (3.50) was valid, Richardson’s extrap- olation was used to improve the results at t = 1. Following [2], we have considered the sequence {Si}iEN, where Si = &(l), with ho = l/40, hk+l = hk/2, lc = 0, 1,2:. . . . Successive transforms of this sequence were then obtained by using the formulae

Tii) = Si,

Tji) = hi+kTi!l - hiTi”_tl”

hi+k - hi , k=1,2 ,...! i=O,l>... .

(4.3)

(4.4)

The results illustrating this extrapolation process are shown in Tables 2-5. Using the relation (3.11), approximate values to u(t), with t > 0, can be obtained. We finish this section by giving an example which illustrates the convergence results for a more general equation (1.3).

Example 4.1. Let o = -0.5 and ,Q = 2. We have set p = 0.7 which implies that v(t) = t0.2 E Vo[O, l] and, consequently, u(t) = t-0.5 E K~a.7 [0, 11. On the other hand, we have p > 1 + ,D and convergence

146 I! Lima, T. Diogo /Applied Numerical Mathematics 24 (1997) 131-148

Table 3 Using the Richardson’s extrapolation to improve the numerical approx- imation of v(l), in Example 4.3

i Tb) 2

Tci) 3

T(“) 4

0 0.22D- 1 0.80D-3 0.1 lD-3 0.28D-4 0.88D-5

1 O.l2D-1 0.28D-3 0.38D-4 O.lOD-4 0.31D-5

2 0.59D-2 O.lOD-3 O.l4D-4 0.35D-5

3 0.30D-2 0.35D-4 0.48D-5

4 O.l5D-2 O.l3D-4

5 0.77D-3

Table 4 Using the Richardson’s extrapolation to improve the numerical approx- imation of w(l), in Example 4.4

i ‘& = Tdi) +) T;(i) TJi) T4(‘)

0 O.l4D-1 0.22D-3 0.84D-6 0.1 ID-8 0.54D-12

1 0.69D-2 0.54D-4 0.1 lD-6 0.63D- 10 0.26D- 13

2 0.35D-2 O.l4D-4 O.lD-7 0.39D-ll

3 O.l8D-2 0.34D-4 O.l7D-8

4 0.88D-3 0.860-6

5 0.44D-3

Table 5 Using the Richardson’s extrapolation to improve the numerical approx- imation of v(l), in Example 4.5

i 3, = T;“’ T,@) T$) T$“’ Tb) 4

0 0.27D- 1 O.lOD-2 0.81D-4 O.llD-4 0.25D-5

1 O.l4D- 1 0.32D-4 0.21D-4 0.32D-5 0.6D-6

2 0.72D-2 0.95D-4 0.54D-5 0.8D-6

3 0.36D-2 0.28D-4 O.l4D-5

4 O.l8D-2 0.8D-5

5 0.92D-3

follows from Theorem 3.2. We have computed the errors in the numerical solution uh to (2.7), for several values of h. This is shown in Table 1, indicating a low order of convergence. This low order could be expected, because in this case the conditions of Theorem 3.4 are not satisfied for any integer m. Therefore there might not exist an error expansion of the type of (3.50) and the Richardson’s extrapolation method is not expected to improve the accuracy of the results.

I? Lima, T Diogo /Applied Numerical Mathematics 24 (1997) 131-148 147

Example 4.2. Let a = -0.5, p = 4 and p = 2.5. Again we have u(t) = t-0.5 but, since z)(t) = t2 E V,“[O, 11, then u E Vt,,, [0, 7’1. The convergence condition p > 1 + p is satisfied and an application of Theorem 3.4, with m’ = 2, gives first order of convergence. This is confirmed by the results displayed in Table 1. The absolute errors of the approximations obtained by the process (4.3)-(4.4), are shown in Table 2. Each column of this table corresponds to a new transform. We see that the convergence is accelerated in the first two steps of the extrapolation process, while in the subsequent steps there is no significant improvement. This is in agreement with the remainder of the asymptotic error expansion (3.50) being of order O(h2).

Example 4.3. Let a = 1 and p = 1.5. Imposing the convergence condition 1-1 > 1 + j3 gives p < 0.5 and we chose ,0 = 0. Note that in this case u(t) = t is an infinitely differentiable function. However v(t) = t E Vp[O, 11. Therefore, an asymptotic expansion of the type of (3.50) is valid with m = 1, which is confirmed by the results displayed in Table 1. An application of the Richardson’s extrapolation process gives the results contained in Table 3. We observe that the convergence is accelerated only from the first to the second column, but not in the subsequent columns.

Example 4.4. Let CI: = 4.5 and p = 1.5. As in the previous example, we chose /3 = 0. However in this case u E V(,[O, I] and u E V,“[O, 11, due to the higher value of a. Therefore, according to Theorem 3.4,

the numerical solution wh has an asymptotic error expansion with an O(h4) remainder term. The absolute errors obtained in this case are displayed in Table 1, indicating that the convergence order is one. In Table 4 we show the extrapolated results. We would expect the convergence to be accelerated at least until the third transform. The errors indicate that the order of convergence is approximately equal to 1 for the first column (initial sequence), 2 for the second column (first transform), 3 for the third column (second transform), 4 for the fourth column (third transform) and 4.4 for the fifth column. The continuation of the extrapolation process has shown that the convergence cannot be accelerated any further.

Example 4.5. Let K(t, s) := etps and

g(t) := t” - f(2 - et (2 + 2t + t2))

in (1.3), with Q = 1.8 and I_L = 1.2. Choosing p = 0, then u(t) = t’.* E V~OIO, l] and v(t) =

t’.* E V,“[O, 11. The values in Table 1 indicate first order of convergence as predicted by Theorem 3.4. Moreover we would expect the error in the numerical solution uh to be O(h). This is in agreement with the results obtained by Richardson’s extrapolation which are shown in Table 5.

Acknowledgements

The authors would like to thank Dr. Tao Tang for helpful suggestions which led to an improvement of the presentation of this paper. The first author is also grateful to Fundacao Luso-Americana para o Desenvolvimento for financial support.

148 F! Lima, 7: Diogo /Applied Numerical Mathematics 24 (1997) 131-148

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