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Large-Time Asymptotics for Solutions of a Generalized

Burgers Equation with Variable Viscosity

By Ch. Srinivasa Rao and Engu Satyanarayana

In this paper, we discuss the large-time asymptotics for periodic solutions of

a generalized Burgers equation with variable viscosity (GBEV). Large-time

asymptotics for the solutions of the GBEV depend on the parameters present

in the partial differential equation and also the period of the solution of the

GBEV. Large-time asymptotic expansions of the solutions are obtained by

improving the solution of the linearized GBEV for certain parametric regions

via a perturbative approach. These constructed large-time asymptotics are

compared with the corresponding numerical solutions and are found to be

in good agreement for large time. For certain other parametric region, our

numerical study suggests that the solution of the inviscid GBEV describes the

large-time behavior of the periodic solutions of the GBEV.

1. Introduction

In this paper, we discuss the large-time asymptotics for solutions of ageneralized Burgers equation with variable viscosity (GBEV), namely,

ut + uux =

(t + 1)M ux x , 0 < x < l, t > 0, (1)

subject to the initial and boundary conditions

u(x , 0) = u0(x ), 0 x l, (2)

Address for correspondence: Dr. Ch. Srinivasa Rao, Department of Mathematics, Indian Institute ofTechnology Madras, Chennai 600036, India; e-mail: [email protected]

DOI: 10.1111/j.1467-9590.2011.00509.x 1STUDIES IN APPLIED MATHEMATICS 0:123C 2011 by the Massachusetts Institute of Technology

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2 Ch. S. Rao and E. Satyanarayana

u(0, t) = u(l, t) = 0, t 0. (3)Here M 0, > 0, andl > 0 are constants. Further, u0(x ) is continuous andtakes the value zero for x = 0 and x = l.

For the parametric regions (i) 0 M < 1 and (ii) M = 1, >l2

2 , thelarge-time asymptotics for solutions of (1)(3) are constructedvia a perturbative

approach assuming that a solution of the linearized equation

ut =

(t + 1)M ux x , 0 < x < l, t > 0 (4)

of (1) satisfying (3) describes the large-time behavior of the solutions of (1)(3).

The large-time asymptotic solutions of (1)(3) constructed here are compared

with the numerical solutions of (1)(3) obtained by a numerical scheme due to

Dawson [1]. They agree very well for large time. The perturbative approach

used for constructing large-time asymptotic solutions is similar to the work ofSachdev et al. [2, 3]. It may be noted that we do not use the initial profile

in the construction of the large-time asymptotic solutions of (1)(3). Our

numerical study shows that the large-time asymptotics of (1)(3) constructed

here describe the large-time behavior of the solutions of (1)(3) for different

initial conditions. However there is a constant, called the old-age constant, in

the constructed asymptotic solutions of (1). This constant needs to be found by

matching with the relevant numerical solution of (1)(3) or by some other

means. It may depend on the parameters present in the partial differential

equation and the initial and boundary conditions. Our numerical study suggests

that the large-time behavior for solutions of (1)(3) for the parametric regionM > 1 is described by the solution x/ of the inviscid generalized Burgers

equation, namely,

u + uux = 0, = t + 1. (5)The parametric region M = 1, l2

2will be dealt elsewhere.

Equation (1) with M = 0 becomes the most celebrated Burgers equationut + uux = ux x , 0 < x < l, t > 0. (6)

The exact periodic solution of (6) subject to (3) is given by

u(x, t) = 4l

n=1

nexp

n

2 2t

l2

sinnx

l

1 + 2

n=1exp

n

2 2t

l2

cos

nxl

(7)

(see Sachdev [10], p. 33). It may be noted that the exact solution (7) of (6) is

obtained from the solution of (6) subject to the boundary conditions (3) and

the initial datum

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Solutions of a Generalized Burgers Equation 3

u(x, 0) = u0 sinx

l

, 0 x l, (8)

under the restriction that is sufficiently small. Here u0 > 0 is a constant.

Expanding the solution (7) of the Burgers equation (6) as a series in descending

exponential functions for sufficiently large time, we obtain large-time asymptotic

expansion for periodic solutions of (6) subject to (3) and (8) as

u(x, t) = 4l

e

2t/ l2 sinx

l

e2 2t/ l2 sin

2x

l

+ e3 2t/ l2

sinx

l

+ sin

3x

l

+

as t . (9)

We show that the large-time asymptotic solution for (1)(3) constructed here (for

the parametric regions 0

M < 1) contains (9) as a special case when M=

0.

A special case of (1) with M = 1/2, namely,

u t + uux =

t + 1 ux x

was derived by Enflo and Rudenko [4] from Khokhlov, Zabolotskaya, and

Kuznetsov (KZK) equation while studying a plane wave in the center of a

bounded nonlinear acoustic beam (see also Enflo and Hedberg [5], p. 215).

Crighton [6] derived generalized Burgers equations (GBEs) of the form

ut + uux = (t)ux x ,where (i) (t) = t + t0, (ii) (t) = exp(t/t0), (iii) (t) = (t0 t)1. TheseGBEs are referred to as cylindrical far-field GBE, spherical far-field GBE, and

exponential horn GBE, respectively. Note that the GBE (iii) corresponds to (1)

with M = 1 (see also Sionoid and Cates [7]). Cates [8] transformed the GBEut + uux = (t)ux x (10)

into

w + ww = (/(1 ))w (11)via the transformation

u(x, t) = (1 + t)1w(x (1 + t)1, 1 (1 + t)1) + x(1 + t)1, (12)

= x1 + t, =

t

1 + t. (13)

An interesting observation of Cates [8] is that the cylindrical far-field GBE

((t)

=t

+1) transforms to the exponential horn GBE

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w + ww =1

1 w

via the transformation (12) and (13). Thus, these GBEs are equivalent.

For a related study, we may refer to [915].Assume that u1(x) is a continuous function on R satisfying the following

conditions:

(i) u1(0) = u1(l) = 0,(ii) u1(x ) is periodic with period 2l,

(iii) u1(x ) is anti-symmetric in the interval ( l, l).Because the GBE (1) and the function u1(x ) are invariant under the

transformations

(i) x

x , u

u,

(ii) x x + 2l, u u,the solution of the GBE (1) subject to the initial profile u1(x ) is given by the

solution of (1) subject to (3) with the initial profile u1(x) restricted to the

interval [0, l]. Because of this reason, without the loss of generality, we may

refer to the solutions of (1)(3) as periodic solutions.

The organization of this paper is as follows. We construct large-time

asymptotic periodic solutions of (1)(3) for the parametric regions

0 M < 1 and M = 1, > l2/ 2 in Sections 2 and 3, respectively. Theconclusions of the present paper are set forth in Section 4.

2. Large-time asymptotics for periodic solutions of (1) when 0 M < 1

In this section, we construct large-time asymptotics for periodic solutions of

(1)(3) via a perturbative approach. The asymptotics of (1)(3) constructed

here are compared with numerical solutions of (1)(3) obtained by Dawsons

[1] numerical scheme for a specific initial condition.

We assume that, for large time, the diffusion term dominates the convection

term of (1) and hence the periodic solution of (1) subject to (2)(3) isasymptotic in the limit t to the old-age solution of (1), namely,

u(x, t) = Ae(t) sinx

l

(14)

when 0 M < 1. Here

(t) = 2

l2(1 M) (t + 1)1M (15)

and A is old-age constant.

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Note that the function given in (14) is a solution of the linear partial

differential equation (4) satisfying the boundary conditions (3).

We seek u(x , t) in the form

u(x, t) = Ae(t)

sinx

l+ (x, t) as t . (16)

Here (x, t) O(e(t)) as t . The correction term (x , t) takes intoaccount the effect of the nonlinear term in (1) for large time. Substitution of

(16) into (1) gives the following partial differential equation for(x , t):

t

(t + 1)M x x + x + Ae(t) sin

xl

x +

A

le(t) cos

xl

= A2

2le2(t) sin

2x

l .(17)

Ignoring the higher order terms, for sufficiently large t, we get

t

(t + 1)M x x A2

2le2(t) sin

2x

l

. (18)

Substituting the particular form

(x , t) f(t)sin

2x

l

(19)

in (18) yields

f(t) + 4 f(t) A2

2le2(t). (20)

The general solution of (20) is given by

f(t) A2

2le4(t)

e2(t)dt + ce4(t), c is the integration constant

A2

2le4(t)

e2(t)dt as t . (21)

Applying integration by parts for the integral in (21), we arrive at

f(t) A2l

4e2(t)

(t + 1)M M

a(t + 1)2M1 + M(2M 1)

a2(t + 1)3M2

M(2M 1)(3M 2)a3

(t + 1)4M3 + (up to [m] terms)

(22)

as t ; here a = 2 2/l2, m = 1/(1 M) and [m] is the greatest integerless than or equal to m. It may be observed that (i) if m is a positive integer,

then (m + 1)th term and the subsequent terms in the bracket of (22) becomezero (i.e., for M

=0, 1/2, 2/3, . . . , the right-hand side (RHS) expression of

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Equation (22) contains only finite number of terms) and (ii) if m is not a

positive integer (i.e., M = 0, 1/2, 2/3, . . .), then those terms with negativepowers may be neglected as t . Therefore, the function for (18) takesthe general form

(x, t)

n=2cne

n2(t) sinnx

l

+ f(t)sin

2x

l

as t . (23)

Here f(t) is as in (22) and in the summation of (23), n has to vary from 2 to because of the requirement that

(x, t) O

e(t)

as t .It follows from (23) that

(x, t) f(t)sin2xl as t . (24)Thus, the large-time asymptotic periodic solution of (1)(3) for 0 M < 1 isgiven by

u(x , t) = Ae(t) sinx

l

+ f(t)sin

2x

l

+ (25)

as t ; here (t) = 2l2(1M) (t + 1)1M and f(t) is as in (22).

Motivated by the form of the asymptotic solution (25) for (1)(3), we seek

the large-time asymptotic periodic solution u of (1)(3) as follows:

u(x , t) = e(t) f1(x, t) + e2(t) f2(x, t) + e3(t) f3(x, t)+ + en(t) fn(x, t) + as t . (26)

Substituting the expression (26) for u into (1) and then equating the

coefficients of en(t), n = 1, 2, 3, . . . to zero yield the following system oflinear second order partial differential equations for the unknown functions fn:

f1,t (t) f1 l2

2(t) f1,x x = 0, (27)

f2,t 2(t) f2 l2

2(t) f2,x x = [ f1 f1,x ], (28)

f3,t 3(t) f3 l2

2(t) f3,x x = [ f1 f2,x + f2 f1,x ], (29)

. . .

fn,t n(t) fn l2

2(t) fn,x x = [ f1 fn1,x + f2 fn2,x + + fn1 f1,x ],

(30). . . .

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The solution f1 of (27) satisfying the relevant boundary conditions

f1(0, t) = f1(l, t) = 0 (31)is given by

f1(x , t) =

n=1Ane

(n21)(t) sinnx

l

A1 sinx

l

as t . (32)

In view of (32) and (28), we have

f2,t 2(t) f2 l2

2(t) f2,x x =

A21

2lsin

2x

l

(33)

as t . Motivated by the RHS expression of (33), let us choose theparticular solution

f2p(x , t) = (t)sin

2x

l

. (34)

Then (t) satisfies

(t) + 2(t)(t) + A21

2l= 0. (35)

Solving (35), we arrive at

(t) A21l

4

(t + 1)M M

a(t + 1)2M1 + M(2M 1)

a2(t + 1)3M2

+ (upto [1/(1 M)] terms)

as t (36)

C1(t + 1)M + C2(t + 1)2M1 + C3(t + 1)3M2 + as t ,(37)where a = 2 2 /l2. The solution f2 of (33) satisfying the relevant boundaryconditions

f2(0, t) = f2(l, t) = 0 (38)is given by

f2(x, t) =

n=2ne

(n22)(t) sinnx

l

+ (t)sin

2x

l

, (39)

where n are constants. In the summation of (39), n cannot start from 1 because

of the requirement that

f2(x , t)e2(t)

O(e(t)) as t

.

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It is easy to see that

f2(x, t) (t)sin

2x

l

as t , (40)

where is given by (36). Making use of (32) and (40), Equation (29) becomes

f3,t 3(t) f3 l2

2(t) f3,x x =

A1

2l(t)

3sin

3x

l

sin

xl

(41)

as t . Motivated by the RHS expression of (41), we attempt the particularsolution of (41) in the form

f3p(x, t) = (t)sin3x

l + (t)sin x

l . (42)Substituting (42) into (41) and comparing the coefficients of sin(3x

l) and

sin( xl

), respectively, we get the following inhomogeneous first order ordinary

differential equations for and , respectively:

+ 6 + 3A12l

(t) = 0, (43)

2 A12l

(t) = 0. (44)

Solving (43) and (44), we arrive at

(t) = 3A12l

e6(t)

(t)e6(t)dt + c1e6(t), (45)

(t) = A12l

e2(t)

(t)e2(t)dt + c2e2(t); (46)

here c1 and c2 are integration constants. In view of (36), we obtain

e6(t) (t) e6(t)dt A

21l

3

24 32(t + 1)2M + O((t + 1)3M1) as t ,

(47)

e2(t)

(t) e2(t)dt A21l

3

8 32(t + 1)2M + O((t + 1)3M1) as t .

(48)

Inspired by Equations (47) and (48), we write the particular solutions for

and as follows:

p(t) = d1(t + 1)2M + d2(t + 1)3M1 + d3(t + 1)4M2 + as t ,(49)

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p(t) = d1(t + 1)2M + d2(t + 1)3M1 + d3(t+ 1)4M2 + as t .(50)

Substituting (49)(50) in (43) and (44), respectively, and solving fordi, d

i,

we arrive at

d1 =C1

a, di =

Ci di1[i M i + 2]a

, i 2, (51)

d1 =C1

a, di =

Ci di1[i M i + 2]a

, i 2, (52)

where

a

=6 2

l2, Ci

= 3A1

2l

Ci , i

1, (53)

a = 22

l2, Ci =

A1

2lCi , i 1, (54)

and Ci are as in (37). In view of Equations (45) and (49),

(t) d1(t + 1)2M + d2(t + 1)3M1 + d3(t + 1)4M2 + as t .(55)

Because

f3(x, t) e3(t)

O(e2(t)

(t+ 1)M

) as t , (56)c2 in (46) must be zero and hence

(t) = A12l

e2(t)

(t)e2(t)dt

d1(t + 1)2M + d2(t + 1)3M1 + d3(t + 1)4M2 + as t .(57)

Then the solution f3 for (41) subject to the relevant boundary conditions

f3(0, t) = f3(l, t) = 0 (58)is given by

f3(x, t) =

n=2kne

(n23)(t) sinnx

l

+ (t)sin

3x

l

+ (t)sin

xl

,

(59)

where kn are constants, and are as in (55) and (57), respectively. Thus,

f3(x , t) (t)sin3x

l + (t)sinx

l as t . (60)

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In the similar way, we can solve the linear partial differential Equation (30) for

fn, where n = 4, 5, . . . .Finally, the large-time asymptotic periodic solution of (1)(3) is given by

u(x , t) = A1e(t) sin

xl

+ e2(t) (t)sin2xl

+ e3(t)

(t)sin

3x

l

+ (t)sin

xl

+ (61)

as t ; here A1 is the old-age constant and the functions , and aregiven by (36), (55), and (57), respectively, and(t) is as in (15).

It is to be noted that we have not used the initial condition (2) anywhere

in the process of constructing the large-time asymptotic periodic solution of

(1)(3). It can be observed that the solution (61) of (1)(3) has exponential

decay as t .It can be seen that the asymptotic solution (61) for M = 0 reduces to

u(x, t) = (A1e)et sinx

l

(A1e

)2

4 / le2t sin

2x

l

+ (A1e)3

(4 / l)2e3t

sin

3x

l

+ sin

xl

+ (62)

as t ; here = 2/l2. If we choose the old-age constant A1 to be 4 l e,the large-time asymptotic solution (62) becomes the asymptotic solution (9) ofthe Burgers equation (6).

For the sake of numerical study, we take M = 0.5. In this case, theasymptotic periodic solution u of (1) satisfying (3) is given by (61) with

(t) = 22

l2

t + 1, (63)

(t) = A21l(l

2 4 2t+ 1)16 32

, (64)

(t) = 3A31

5l

6

432 42+ 5l

4

t + 136 2

l2

3(t + 1)

16 22, (65)

(t) =A31

l6

16 42+ l

4

t+ 14 2

+ l2(t + 1)

16

2

2. (66)

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0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

x

u(x,

t)

0 0.5 1 1.5 2 2.5 3 3.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x

u(x,

t)

Figure 1. [Left] Initial profile given in (67) at t = 0; [right] numerical solution (dashed) of(1) subject to (67) and (68), asymptotic solution (61) (solid), and the linear solution (69)

(dashdotted) with M

=0.5,

=0.5 at t

=5.

We now solve, numerically, the variable coefficient Burgers equation (1) with

M = 0.5 and = 0.5 subject to the initialboundary conditions

u(x, 0) = sinx , x [0, ], (67)

u(0, t) = u(, t) = 0, t 0. (68)

We chose t = 0.0001, x = 0.0062 and used Dawsons [1] scheme forfinding the numerical solution of the GBE (1) with the initial and boundary

conditions (67) and (68). We chose l = for the sake of simplicity. As theinitial profile evolves in time under the generalized Burgers equation (1),

the solution of (1) for sufficiently large time behaves like the solution (14)

of the linearized equation (4) for 0 M < 1. We computed umaxsin(xmax)

e2

t+1,where xmax is the value of x when u attains its maximum value umax on (0,

) at different times and chose the converged value for A1. Convergence

of A1 means that the nonlinear terms are negligible, that is, we reached

the linear regime. Here A1 is 2.06. The agreement between numerical and

asymptotic solutions is very good. Figures 13 show the asymptotic solution

(61) with (t), (t), (t), and (t) given by (63)(66), numerical solution of

the variable coefficient Burgers equation (1) subject to (67) and (68) with

M = 0.5 and = 0.5 and the solution

uli n(x , t) = A1e2

t+1 sinx (69)

of linearized form (4) at various times. We observe that our asymptotic solution

(61) starts agreeing with the relevant numerical solution much before the linear

regime.

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0 0.5 1 1.5 2 2.5 3 3.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x

u(x,

t)

0 0.5 1 1.5 2 2.5 3 3.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x

u(x,

t)

Figure 2. Numerical solution (dashed) of (1) subject to (67) and (68), asymptotic solution (61)

(solid), and the linear solution (69) (dashdotted) with M = 0.5, = 0.5 at t = 10, 15,respectively.

0 0.5 1 1.5 2 2.5 3 3.50

0.005

0.01

0.015

0.02

0.025

x

u(x,

t)

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

x

u(x,

t)

Figure 3. Numerical solution (dashed) of (1) subject to (67) and (68), asymptotic solution (61)

(solid), and the linear solution (69) (dashdotted) with M = 0.5, = 0.5 at t = 20, 50,respectively.

3. Large-time asymptotics for periodic solutions of (1)

when M = 1 and > l2/ 2

In this section, we construct large-time asymptotics for periodic solutions

of (1)(3) for M = 1 and > l2/ 2 via the perturbative approach as inSection 2. The large-time asymptotics of (1) constructed here are compared

with numerical solutions of (1)(3) for specific initial conditions.

The initialboundary value problem (1)(3) with M = 1 becomes

ut + uux =

t + 1 ux x , 0 < x < l, t > 0, (70)

u(x, 0)

=u0(x), 0

x

l, (71)

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u(0, t) = u(l, t) = 0, t 0. (72)

We assume that, for large time, the convection term gets dominated by the

diffusion term in (70) and then the periodic solution of (70) subject to (71) and

(72) is asymptotic to function

u(x , t) = A0(t + 1) 2/ l2 sinx

l

(73)

for > l2/ 2. Here A0 is old-age constant. Note that the function given in

(73) is a solution of the linearized partial differential equation

ut =

t + 1 ux x , 0 < x < l (74)

of (70) subject to the initial and boundary conditions (71) and (72). We seek

the asymptotic periodic solution u of (70) in the form

u(x, t) = A0(t + 1)k sinx

l

+ (x , t) as t , (75)

where k = 2/l2 and(x , t) O((t + 1)k) as t . Substitution of (75)into (70) gives the following partial differential equation for(x , t):

t

t + 1 x x + x + A0(t + 1)k sin

xl

x +

A0

l(t + 1)k cos

xl

= A20

2l

(t

+1)2k sin2x

l . (76)

Ignoring the higher order terms in (76), we get

t

t + 1 x x A20

2l(t + 1)2k sin

2x

l

(77)

as t . Let

(x, t) = f(t)sin

2x

l

as t . (78)

Substitution of the expression for given in (78) into (77) yields

f(t) + 4k f(t)t+ 1 =

A20

2l(t + 1)2k. (79)

Solving (79) for f, we arrive at

f(t) = c1(t + 1)4k A20

2l(1 + 2k) (t + 1)12k, (80)

where c1 is the integration constant. Then the correction term satisfying the

relevant boundary conditions

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(0, t) = (l, t) = 0 (81)is given by

(x, t) =

n=2cn(t + 1)n

2

k sinnx

l

+

c1(t + 1)4k A20

2l(1 + 2k) (t + 1)12k

sin

2x

l

, (82)

where cn are constants and the summation in (82) started from n = 2 becauseof our assumption that

(x, t) O((t + 1)k) as t . (83)It follows from (82) that

(x , t) A20

2l(1 + 2k) (t + 1)12k sin

2x

l

as t . (84)

The condition on (see (83)) is satisfied because of our assumption that k >

1. Therefore, the asymptotic periodic solution u of (70)(72) is

u(x , t) = A0(t + 1)k sinx

l

B(t + 1)(2k1) sin

2x

l

+ (85)

as t

, where B=

A20

2l(1+2k)and k

= 2/ l2.

Motivated by the form of asymptotic solution (85) for (70) satisfying (72),

we seek the large-time asymptotic periodic solution u of (70) subject to (71)

and (72) as follows:

u(x , t) = (t + 1)kf1(x, t) + (t + 1)2kf2(x, t)+ (t+ 1)3kf3(x, t) + + (t + 1)nkfn(x, t) + (86)

as t and k = 2/l2. Substituting the expression (86) for u into (70)and equating the coefficients of (t + 1)nk, n = 1, 2, 3, . . . to zero yield thefollowing system of linear second order partial differential equations for fn:

(t + 1) f1,t k f1 f1,x x = 0, (87)

(t + 1) f2,t 2k f2 f2,x x = (t + 1)[ f1 f1,x ], (88)

(t + 1) f3,t 3k f3 f3,x x = (t + 1)[ f1 f2,x + f2 f1,x ], (89)

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(t + 1) fn,t nk fn fn,x x = (t + 1)[ f1 fn1,x + f2 fn2,x + + fn1 f1,x ],(90)

The general solution f1 of (87) subject to the relevant boundary conditions

f1(0, t) = f1(l, t) = 0 (91)is given by

f1(x, t) =

n=1An(t + 1)(n

21)k sinnx

l

A1 sinx

l

as t . (92)

In view of (92), Equation (88) reduces to

(t + 1) f2,t 2k f2 f2,x x = A21

2l(t + 1) sin

2x

l

(93)

as t . Motivated by RHS expression of (93), we seek the particularsolution f2p for (93) as

f2p(x, t) = (t + 1)g(x ). (94)Because of (72), the boundary conditions for f2p are

f2p(0, t) = f2p(l, t) = 0. (95)Then (94) gives

g(0) = g(l) = 0. (96)Substitution of the expression for f2p given in (94) into (93) yields

g +

2k 1

g = A

21

2lsin

2x

l

. (97)


g(x ) = c3 cos2k 1 x+ c4 sin2k 1 x A2

1

2l(1 + 2k) sin2x

l

.

(98)

Here c3 and c4 are constants. The boundary condition (96)1 gives c3 = 0. Thisresult along with the condition (96)2 and (98) gives

eitherc4 = 0 or1

k= 2 n2, n is any integer. (99)

Recall that k = 2l2

> 1. This, in turn, implies that 0 < 1k

< 1 and hence there

exists no integer satisfying (99)2. Hence, we must choose c4=

0 and the

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general form of the solution f2 of (93) is

f2(x , t) =

n=2Bn(t + 1)(n22)k sin

nx

l A

21

2l(1 + 2k) (t + 1) sin

2x

l

, (100)

where Bn are constants. In the summation of (100), n varies from 2 to because of the requirement that

(t + 1)2kf2(x, t) O((t + 1)k) as t .Retaining the dominant term in (100), we get

f2(x, t) B(t + 1) sin2xl as t , (101)where B = A21

2l(1+2k) . Note that (t + 1)2kf2(x, t) O((t + 1)k) as t ,because of the condition k > 1. Making use of (92) and (101), Equation (89)

becomes

(t + 1) f3,t 3k f3 f3,x x = D(t + 1)2

3sin

3x

l

sin

xl

(102)

as t

. Here D

=A1B

2l. The RHS expression of (102) suggests the

particular solution f3p to be of the form

f3p(x, t) = (t + 1)2h(x ). (103)Substitution of the expression (103) into (102) leads to the second order linear

ordinary differential equation

h +

3k 2

h =

D

3sin

3x

l

sin

xl

. (104)


h(x) = c5 cos

3k 2

x

+ c6 sin

3k 2

x

+D

2(1 k) sinx

l

3

D

2(1 + 3k) sin

3x

l

, (105)

where c5 andc6 are constants. Because of (72), the relevant boundary conditions

for f3p are

f3p(0, t)

=f3p(l, t)

=0. (106)

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Then (103) gives

h(0) = h(l) = 0. (107)

The boundary conditions (107) yield c5 = 0 and

either c6 = 0 or2

k= 3 n2, n is any integer. (108)

As 0 < 2k

< 2, there exists no integer satisfying (108)2. Thus, c6 = 0. Hence,the general solution for f3 is

f3(x , t) =

n=2Cn(t + 1)(n23)k sin

nx

l +

D

2(t + 1)2

1

1 k sinx

l

3

1 + 3k sin

3x

l

, (109)

where Cn are constants. In the summation of (109), n varies from 2 to because of the assumption that

(t + 1)3kf3(x , t) O((t + 1)(2k1)) as t .

It follows from (109) that

f3(x , t) D

2(t + 1)2

1

1 k sinx

l

3

1 + 3k sin

3x

l

(110)

as t . In the similar way, we can solve the linear partial differentialEquation (90) for fn, where n = 4, 5, . . .

Thus, the large-time asymptotics for periodic solution u of (70)(72) is

u(x , t) = A1(t + 1)k sin

x

l + B(t + 1)(2k1) sin

2x

l

+ D2

(t + 1)(3k2)

1

1 k sin

x

l

3

1 + 3k sin

3x

l

+ as t , (111)

where k = 2/ l2, B = A212l(1+2k) , D = A1B2l and A1 is the old-age constant.

It may be noted that the initial condition (71) is not used while constructing

large-time asymptotics for periodic solutions of (70). It may be observed that

the large-time asymptotic solution u given in (111) for (70)(72) has algebraic

decay for sufficiently large t.

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We have solved numerically the variable coefficient Burgers equation (70)

with = 1.2 subject to the initial and boundary conditions:

u(x, 0) =

2x

, if 0 x

2 ,

2

1 x

, if

2 x ,

(112)

u(x, 0)

=

4x

, if 0 x

4,

4

3

1 x

, if

4 x

2,

4x

3 , if

2 x 3

4 ,

4

1 x

, if

3

4 x ,

(113)

u(x, 0) = sinx, x [0, ], (114)

u(0, t) = u(, t) = 0, t 0. (115)

As in Section 2, we used Dawsons [1] scheme for finding the numerical

solutions of the GBE (70) satisfying the boundary conditions (115) and the

initial conditions (112), (113), and (114), respectively.

We computed umaxsin(xmax)

(t+ 1) , where xmax is the value ofx when u attains itsmaximum value umax on (0, ) at different times and chose the converged

value for A1. The values of the old-age constant A1 corresponding to the

initial profiles (112), (113), and (114) are 0.784, 0.944, and 0.953, respectively.

The agreement between numerical and asymptotic solutions is quite good forall the three initial profiles when t is large. For the sake of illustration, we

present below in Figures 47 the comparison of the asymptotic solution (111),

numerical solution of (70) subject to (113) and (115), and the solution of the

linearized equation of (70) given by

u(x, t) = A1(t + 1) sinx (116)

when

=1.2 at different times.

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0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

u(x,

t)

Figure 4. [Left] Initial profile given by (113) at t = 0; [right] numerical solution (dashed) of(70) subject to (113) and (115), asymptotic solution (111) (solid), and the linear solution

(116) (dashdotted) with = 1.2 at time t = 0.1.

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

u(x,

t)

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

u(x,

t)

Figure 5. Numerical solution (dashed) of (70) subject to (113) and (115), asymptotic solution

(111) (solid), and the linear solution (116) (dashdotted) with = 1.2 at times t = 0.2, 0.5,respectively.

0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x

u(x,

t)

0 0.5 1 1.5 2 2.5 3 3.50

0.01

0.02

0.03

0.04

0.05

0.06

x

u(x,

t)


(111) (solid), and the linear solution (116) (dashdotted) with = 1.2 at times t = 1, 10,respectively.

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0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

4x 10

x

u(x,

t)

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

x

u(x,

t)


(111) (solid), and the linear solution (116) (dashdotted) with = 1.2 at times t = 100, 300,respectively.

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

x

u(x,

t)

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

u(x,

t)

Figure 8. [Left] Initial profile given by (114) at t = 0; [right] numerical solution (dashed) of(1) subject to (114), (115) with M = 1.5, = 0.5 at time t = 5. Solid profile corresponds tothe inviscid solution (117) of (1) at t = 5.

Remark. Based on our numerical study, we expect that the large-time

asymptotic behavior of the solutions of (1)(3) on [0, l] for M > 1 is given by

the solution

u(x, t) = x /(t + 1) as t (117)of the inviscid equation, namely,

u + uux = 0, (118)where = t + 1. Figures 8 and 9 show the evolution of the initial profile(114) under the GBE (1) for M = 1.5, = 0.5 to the inviscid solution (117).

Remark. Parker [16] studied an initial value problem posed for a GBE

on infinite interval. He made use of ColeHopf like transformation and then

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0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

u(x,

t)

0 0.5 1 1.5 2 2.5 3 3.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

x

u(x,

t)

Figure 9. Numerical solution (dashed) of (1) subject to (114), (115) with M = 1.5, = 0.5at times t = 10, 50, respectively. Solid profiles correspond to the inviscid solution (117) of (1)at t

=10, 50, respectively.

linearized the resulting differential equation as well as the resulting initial

condition. We deal with the resulting initial condition as it is without linearizing

it unlike in Parkers [16] work. Following Parkers [16] approach, it is easy to

arrive at the approximate solution of the GBE (1) subject to the initial profile

(8) and the boundary conditions (3) as

u(x, t) = 2 wx (x , T)1 (t + 1)M(1 w(x, T)) , (119)

where

w(x, T) c0 +

j=1cj exp((j )2T/l2)cos

jx

l

, (120)

c0 =1

l

l0

exp

k

coss

l

1

ds,

= ekI0(k), (121)

cj = 2l

l

0

exp

k

coss

l

1

cos

j s

l

ds,

= 2ekIj (k). (122)

Here T = 1M((1 + t)1M 1), k = lu 02 and In(x)(= 1

0

ex cos cos(n )d )

are the modified Bessel functions of the first kind satisfying

x 2y + x y (x 2 + n2)y = 0.

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It is easy to see that for M = 0, the solution (119) reduces to

up(x, t) = 2wx (x, t)

w(x, t). (123)

In fact, the solution up(x , t) given in (123) (see (120)(122)) is the known

exact periodic solution (see Sachdev [10], p.32) of the Burgers equation (6)

subject to (8) and (3).

Our numerical study showed that this approximate solution (119)(122)

agrees well with the relevant numerical solution forM sufficiently small.

4. Conclusions

In this paper, we have studied the solutions of a GBEV (1) for large time. The

large-time asymptotics for periodic solutions of (1)(3) have been constructed

by a perturbative approach in the manner of Sachdev et al. [2, 3] for 0 M l2/ 2. Large-time asymptotic expansionsobtained in the parametric regions 0 M < 1 and M = 1, > l2/ 2 forthe solutions of (1)(3) are compared with relevant numerical solutions of

(1)(3) for specific initial conditions and are found to be in good agreement.

Our numerical study suggests that the large-time behavior of solutions for the

initialboundary value problem (1)(3) is given by the solution of the inviscid

GBE for the parametric region M > 1. The parametric region M = 1, l2/ 2 will be dealt elsewhere.

References

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Large-time asymptotics for periodic solutions of some generalized Burgers equations,

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INDIAN I NSTITUTE OF TECHNOLOGY MADRAS

(Received June 22, 2010)

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