Applied Mathematics and Computation 164 (2005) 857–864

www.elsevier.com/locate/amc

A decomposition method for findingsolitary and periodic solutions for a

coupled higher-dimensional Burgers equations

Dogan Kaya *, Asif Yokus

Department of Mathematics, Firat University, Matematik Bolumu, Elazig 23119, Turkey

Abstract

In this paper, we consider coupled higher-dimensional Burgers (chdBur) equations.

We find periodic solutions to chdBur equations using a modified Adomian�s decompo-

sition method (mADM). We find both exact and numerical solutions. We compared the

numerical solutions with corresponding analytical solutions. We also show the effective-

ness of the method.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Coupled higher-dimensional Burgers equations; Modified Adomian decomposition

method; Exact periodic solutions; Numerical solutions

1. Introduction

It is well known that there are many nonlinear partial equations in the study

of physics, mechanics and biologics. The solution of these equations can makeauthors know deeply the described process. But because of the complexity of

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.06.012

* Corresponding author.

E-mail addresses: dkaya@firat.edu.tr, dkaya36@yahoo.com (D. Kaya).

858 D. Kaya, A. Yokus / Appl. Math. Comput. 164 (2005) 857–864

nonlinear partial differential equations and the limitations of mathematics

methods, it is difficult for us to get the exact solutions for the problems. In

the recent decades, there has been great development in the exact solutions

for nonlinear partial equations. Up to now, there exist many methods of con-

structing exact solutions, for instance, inverse scattering method [1], Backlund

transformation method [2,3], Lie group method [4], Adomian�s decompositionmethod [5–16]. Of the above methods, Adomian�s decomposition method is a

powerful tool for finding the exact solutions of nonlinear partial equations.

The method is easy and has a strong operability. In this paper, we will study

the coupled higher-dimensional Burgers equations [17,18]

ut ¼ D2uþ 2uux þ 2vuy ; vt ¼ D2vþ 2uvx þ 2vvy ; ð1Þ

where D is the two-dimensional Laplacian operator, i.e.

D2 ¼ o2

ox2þ o2

oy2:

The initial conditions are given as u(x,y, 0) = f1(x,y) and v(x,y, 0) = g1(x,y).The main aim of this paper is to present the ADM for solving the chdBur equa-

tions for different initial conditions and coefficients.

2. Methodology of the mADM

In this section, we outline the main steps of our method. Following the ana-

lysis of Adomian�s decomposition method [6,8,16], we define the linear opera-tors

Lt ¼o

ot; Lxx ¼

o2

ox2; Lyy ¼

o2

oy2:

System (1) is rewritten in terms of the operators as

Ltu ¼ Lxx þ Lyy

� �uþ 2uux þ 2vuy ;

Ltv ¼ Lxx þ Lyy

� �vþ 2uvx þ 2vvy :

ð2Þ

We define inverse operators L�1t which are simply onefold integration operators

in this case, defined by L�1t ¼

R t0ð�Þdt. We apply the inverse operator to both

sides of the system (2), we get

u ¼ f1ðx; yÞ þ L�1t Lxx þ Lyy

� �uþ 2uux þ 2vuy

� �;

v ¼ g1ðx; yÞ þ L�1t Lxx þ Lyy

� �vþ 2uvx þ 2vvy

� �;

D. Kaya, A. Yokus / Appl. Math. Comput. 164 (2005) 857–864 859

which we can write in the following form:

u ¼ f1ðx; yÞ þ L�1t ½ðLxx þ LyyÞuþ 2M1 þ 2N 1�;

v ¼ g1ðx; yÞ þ L�1t ½ðLxx þ LyyÞvþ 2M2 þ 2N 2�;

where M1(u) = uux, N1(u,v) = vuy, M2(u,v) = uvx and N2(v,v) = vvy are the non-

linear terms. The ADM [6,8,16] consists of representing u(x,y, t) and v(x,y, t) in

the decomposition form given by

uðx; y; tÞ ¼X1n¼0

unðx; y; tÞ; vðx; y; tÞ ¼X1n¼0

vnðx; y; tÞ; ð3Þ

respectively. The components un(x,y, t) and vn(x,y, t); n P 0 can be determined

easily in a recursive manner. The nonlinear operators Mi(u,v), Ni(u,v), i = 1, 2

can be defined by the infinite series of Adomian�s polynomials [6–16]:

M1ðu; vÞ ¼X1n¼0

An; N 1ðu; vÞ ¼X1n¼0

Bn;

and

M2ðu; vÞ ¼X1n¼0

Cn; N 2ðu; vÞ ¼X1n¼0

Dn:

Specific algorithms were set in [6,16] for calculating Adomian�s polynomials for

nonlinear terms

Anðu0; . . . ; un; v0; . . . ; vnÞ ¼1

n!dn

dknM

Xnk¼0

kkuk;Xnk¼0

kkvk

!" #k¼0

; nP 0:

ð4ÞSimilarly for the three Adomian�s polynomials can be constructed Bn, Cn and

Dn. The modified decomposition methodology defines [16] the components unand vn for n P 0 by the following recursive relationships:

u0 ¼ 0:0; v0 ¼ 0:0;

u1 ¼ f1ðx; yÞ þ L�1t ½ðLxx þ LyyÞu0 þ 2A0 þ 2B0�;

v1 ¼ g1ðx; yÞ þ L�1t ½ðLxx þ LyyÞv0 þ 2C0 þ 2D0�;

unþ1 ¼ L�1t ½ðLxx þ LyyÞun þ 2An þ 2Bn�;

vnþ1 ¼ L�1t ½ðLxx þ LyyÞvn þ 2Cn þ 2Dn�; nP 1:

ð5Þ

This will enable us to determine the components un, vn recurrently. However, in

many cases the exact solution in a closed form may be obtained. For numerical

comparisons purposes, we construct the solution u(x,y, t) and v(x,y, t)

limn!1

wn ¼ uðx; y; tÞ; limn!1

un ¼ vðx; y; tÞ; ð6Þ

where wnðx; y; tÞ ¼Pn�1

k¼0ukðx; y; tÞ, unðx; y; tÞ ¼Pn�1

k¼0vkðx; y; tÞ, n P 0 and therecurrence relation is given as in (7).

860 D. Kaya, A. Yokus / Appl. Math. Comput. 164 (2005) 857–864

3. The model problems

In first example, we will consider the chdBur equations (1) with the follo-

wing initial conditions

uðx; y; 0Þ ¼ f � S1 tanhðRnÞ; vðx; y; 0Þ ¼ F þ S2 tanhðRnÞ; ð7Þwhere S1 ¼ aF

ffiffiffiffiffiffiffi�d

p, S2 = af + b, R ¼

ffiffiffiffiffiffiffi�d

p, n = kx + ly + c, k = �aF, l = af + b

and f, F, a, b, c, d are arbitrary constants. Again, to find the solution of this

equation, we simply take the equation in an operator form exactly in the same

manner as the form of Eq. (2). We could find the terms of the series as setting

the zeroth components of u0 = 0.0, v0 = 0.0, using the initial conditions (7) with

recurrence relations (5) to find u1, v1, and obtained in succession u2, v2, u3, v3etc. by using (5) with (4) to determine the other individual terms of the decom-position series

u0 ¼ 0; v0 ¼ 0; ð8Þ

u1 ¼ f � S1 tanhðRnÞ; v1 ¼ F þ S2 tanhðRnÞ; ð9Þ

u2 ¼ tð2k2R2S1 sech2ðRnÞ tanhðRnÞ þ 2l2R2S1 sech

2ðRnÞ tanhðRnÞÞ; ð10Þ

v2 ¼ t �2k2R2S2 sech2ðRnÞ tanhðRnÞ � 2l2R2S2 sech

2ðRnÞ tanhðRnÞ� �

;

ð11Þ

u3 ¼ 2kRS1t sech3ðRnÞ½�f coshðRnÞ þ S1 sinhðRnÞ

þ ðk2 þ l2Þ2R4S1t2 sech5ðRnÞð�11 sinhðRnÞÞ þ sinhð3RnÞ�; ð12Þ

v3 ¼ 2lRS2t sech3ðRnÞ½F coshðRnÞ þ S2 sinhðRnÞ

þ ðk2 þ l2Þ2R4S2t2 sech5ðRnÞ11 sinhðRnÞ � sinhð3RnÞ�; ð13Þ

and so on. In this manner the rest of components of the decomposition serieswere obtained. Substituting u0 and (8)–(13) into (6) gives the solution u(x,y, t)

in a series form and the series can be written in a closed form solution by

uðx; y; tÞ ¼ f � S1 tanhðRðkxþ ly þ wt þ cÞÞ;vðx; y; tÞ ¼ F þ S2 tanhðRðkxþ ly þ wt þ cÞÞ;

ð14Þ

which are the solitary wave solutions for Eq. (1) S1 ¼ aFffiffiffiffiffiffiffi�d

p, S2 = af + b,

R ¼ffiffiffiffiffiffiffi�d

p, k = �aF, l = af + b, w = 2bF and f, F, a, b, c, d are arbitrary con-

stants. This result can be verified through substitution [18].

In the second example, we will consider the chdBur equations (1) with the

initial conditions for numerical comparison purpose as

uðx; y; 0Þ ¼ f � S1 cothðRnÞ; vðx; y; 0Þ ¼ F þ S2 cothðRnÞ; ð15Þ

D. Kaya, A. Yokus / Appl. Math. Comput. 164 (2005) 857–864 861

where S1 ¼ aFffiffiffiffiffiffiffi�d

p, S2 = af + b, R ¼

ffiffiffiffiffiffiffi�d

p, n = kx + ly + c, k = �aF, l = af + b

and f, F, a, b, c, d are arbitrary constants.

Again, to find the solution of this equation, we substitute in the scheme (5)

u0 = 0, v0 = 0 and obtained in succession terms by using (5) with (4) to deter-

mine the individual terms of the series, we find as

u0 ¼ 0; v0 ¼ 0;

u1 ¼ f1ðx; yÞ þZ t

0

½ðLxx þ LyyÞu0 þ 2A0 þ 2B0�dt;

v1 ¼ g1ðx; yÞ þZ t

0

½ðLxx þ LyyÞv0 þ 2C0 þ 2D0�dt;

..

.

unþ1 ¼Z t

0

½ðLxx þ LyyÞun þ 2An þ 2Bn�dt;

vnþ1 ¼Z t

0

½ðLxx þ LyyÞvn þ 2Cn þ 2Dn�dt; nP 1;

ð16Þ

where the Adomian polynomials An, Bn, Cn, Dn are given same as in the first

example. Performing the calculations in (5) with (4) using Mathematica and

substituting into (6) gives the exact solution

uðx; y; tÞ ¼ f � S1 cothðRðkxþ ly þ wt þ cÞÞ;vðx; y; tÞ ¼ F þ S2 cothðRðkxþ ly þ wt þ cÞÞ;

ð17Þ

which are the solitary wave solutions for Eq. (1) S1 ¼ aFffiffiffiffiffiffiffi�d

p, S2 = af + b,

R ¼ffiffiffiffiffiffiffi�d

p, k = �aF, l = af + b, w = 2bF and f, F, a, b, c, d are arbitrary con-

stants. This result can be verified through substitution [18].

4. Numerical evaluation and discussions

In order to verify numerically whether the proposed methodology leads to

higher accuracy, we can evaluate the approximate solution using the n-term

approximation (6). It is to be noted that /n and un show clearly the conver-

gence to the correct limit are presented. We achieved a very good approxima-

tion with the actual solution of the equations by using n = 5 terms only of thedecomposition series derived above. It is evident that even using few terms of

series, the overall results getting very close to exact solution, errors can be

made smaller by adding new terms of the decomposition series.

In Table 1, the computational results are obtained for the approximate solu-

tion of Eq. (1) with initial conditions (7). Table 2 presents the numerical results

of the same equation (1) with initial conditions (15) by using the ADM.

Table 1

The numerical results with y = 0.1 when f = 0.1 and F = 0.05 for the solution of Eq. (1) with initial

conditions (7) for the values of a = 0.1, b = 0.01, c = 15.0, d = �0.01

xijti 0.1 0.2 0.3 0.4 0.5

ju�/nj0.1 1.62887E�07 1.27996E�06 4.31493E�05 1.03345E�05 2.04559E�05

0.2 1.62901E�07 1.28006E�06 4.31528E�05 1.03353E�05 2.04573E�05

0.3 1.62914E�07 1.28016E�06 4.31562E�05 1.03368E�05 2.04588E�05

0.4 1.62927E�07 1.28027E�06 4.31596E�05 1.03368E�05 2.04602E�05

0.5 1.62940E�07 1.28037E�06 4.31630E�05 1.03376E�05 2.04616E�05

jv�unj0.1 1.63519E�06 3.17875E�05 1.27816E�04 3.28876E�04 6.75572E�04

0.2 1.63529E�06 3.17899E�05 1.27826E�04 3.28900E�04 6.75619E�04

0.3 1.63538E�06 3.17923E�05 1.27836E�04 3.28924E�04 6.75667E�04

0.4 1.63548E�06 3.17947E�05 1.27845E�04 3.28949E�04 6.75714E�04

0.5 1.63558E�06 3.17972E�05 1.27855E�04 3.28973E�04 6.75761E�04

Table 2

The numerical results with y = 0.1 when f = 0.1 and F = 0.05 for the solution of Eq. (1) with initial

conditions (15) for the values of a = 0.1, b = 0.01, c = 15.0, d = �0.01

xijti 0.1 0.2 0.3 0.4 0.5

ju�/nj0.1 2.40866E�07 1.91284E�06 6.73765E�06 1.82928E�05 5.09595E�05

0.2 2.40895E�07 1.91307E�06 6.73849E�06 1.82953E�05 5.09681E�05

0.3 2.40924E�07 1.91330E�06 6.73932E�06 1.82977E�05 5.09768E�05

0.4 2.40953E�07 1.91353E�06 6.74016E�06 1.83002E�05 5.09854E�05

0.5 2.40982E�07 1.91376E�06 6.74099E�06 1.83026E�05 5.09940E�05

jv�unj0.1 3.34848E�06 5.11615E�05 2.07716E�04 5.96104E�04 1.71293E�04

0.2 3.34892E�06 5.11678E�05 2.07742E�04 5.96185E�04 1.71322E�04

0.3 3.34937E�06 5.11742E�05 2.07768E�04 5.96266E�04 1.71352E�04

0.4 3.34981E�06 5.11742E�05 2.07794E�04 5.96346E�04 1.71381E�04

0.5 3.35026E�06 5.11869E�05 2.07820E�04 5.96427E�04 1.71411E�04

862 D. Kaya, A. Yokus / Appl. Math. Comput. 164 (2005) 857–864

Actually, the present method provides smooth, stable numerical solutions by

using six terms or even using few terms of the decomposition. The accuracyof the ADM algorithm and the chdBur equations (1) with two different initial

conditions (7) and (15) is demonstrated for the absolute errors of Eq. (1) with

their exact solutions.

We can see from Tables 1 and 2, for the chdBur equations (1) when taking

the initial conditions (15) absolute numerical differences are better than when

taking the initial conditions (5). Maybe, it is the nature of the expansion of

the functions to Taylor series. Over all, it is to be noted that five terms only

D. Kaya, A. Yokus / Appl. Math. Comput. 164 (2005) 857–864 863

were used in evaluating the approximate solutions. We achieved a very good

approximation with the actual solution of the equations by using five terms

only of the decomposition derived above. It is evident that the overall errors

can be made smaller by adding new terms of the decomposition series.

5. Conclusions and remarks

In this paper, the ADM was used for the chdBur equations (1) with initial

conditions. The approximate solutions to the equations have been calculated

by using the ADM without any need to a transformation techniques and line-

arization or perturbation of the equations.

In closing, the ADM avoids the difficulties and massive computational work

by determining analytic solutions of the nonlinear equations. We compare theapproximation solution of (6) with the exact solution of the corresponding

equation. Numerical approximations show a high degree of accuracy and in

most cases /n and un, the n-term approximation is accurate for quite low val-

ues of n. The solutions are very rapidly convergent by utilizing the ADM. The

numerical results we obtained justify the advantage of this methodology, even

in the few terms approximation is accurate.

A clear conclusion can be drawn from the numerical results that the ADM

algorithm provides highly accurate numerical solutions without spatial discret-izations for the nonlinear partial differential equations. It is also worth noting

that the advantage of the decomposition methodology displays a fast conver-

gence of the solutions. The illustrations show the dependence of the rapid con-

vergence depend on the character and behavior of the solutions just as in a

closed form solutions.

Finally, we point out that, for given equations with initial values, the corre-

sponding analytical and numerical solutions are obtained according to the recur-

rence equations (5) with (4) usingMathematica. For more implementation of thedecomposition method, one can look at Refs. [19–24] and references therein.

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