The solution of coupled Burgers’ equations using Adomian–Pade technique

Applied Mathematics and Computation 189 (2007) 1034–1047 www.elsevier.com/locate/amc

The solution of coupled Burgers’ equations using Adomian–Pade technique Mehdi Dehghan *, Asgar Hamidi, Mohammad Shakourifar Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15914, Iran

Abstract The purpose of this study is to implement ADM–PADE (MADM–PADE) technique, which is a combination of Adomian decomposition method (modified Adomian decomposition method) and Pade approximation, for solving homogeneous (inhomogeneous) two dimensional parabolic equation. Also, we apply ADM–PADE technique to solve coupled Burgers’ equations. The results obtained by ADM–PADE (MADM–PADE) techniques are compared to those obtained by using ADM (MADM) alone. The numerical results demonstrate that ADM–PADE (MADM–PADE) technique gives the approximate solution with faster convergence rate and higher accuracy than using ADM (MADM). Ó 2006 Elsevier Inc. All rights reserved. Keywords: Adomian Decomposition Method (ADM); Modified Adomian Decomposition Method (MADM); Pade approximation; Heat equation; Coupled Burgers’ equations

1. Introduction In various fields of science and engineering, many physical problems can be described by linear or nonlinear parabolic equations. The Adomian decomposition scheme is a method for solving a wide range of problems whose mathematical models yield equation or system of equations involving differential, partial differential [1]. In recent years, a lot of attention has been focused on using ADM to various types of diffusion equations [2–4]. In this paper we investigate solutions of coupled Burgers’ equations. Burgers’ equation has been found to describe various kinds of phenomena such as mathematical model of turbulence [5] and the approximate theory of flow through a shock wave travelling in viscous fluid [7]. The coupled Burgers’ system was derived by Esipov [8]. It is a simple model of sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid suspensions or colloids, under the effect of gravity [9]. Using the Hopf–Cole transformation, Fletcher [10] gave an analytical solution for the system of two dimensional Burgers’ equations. Several numerical methods for solving this equation have been given such as algorithms based on the cubic spline function technique [11], the explicit–implicit method [12], and the implicit finite-difference scheme [13]. Soliman [14] *

Corresponding author. E-mail addresses: [email protected] (M. Dehghan), [email protected] (A. Hamidi), [email protected] (M. Shakourifar).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.179

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

1035

used the similarity reductions for the partial differential equations to develop a scheme for solving Burgers equation. High-order accurate schemes for solving the two dimensional (2D) Burgers equations have been used in [15,16]. The fourth-order accurate two-point compact scheme and the fourth-order accurate Dufort–Frankel scheme have been discussed in [16]. The variational iteration method was used to solve the one dimensional (1D) Burgers’ and coupled Burgers’ equations [17], the solution was obtained under a series of initial conditions and was transformed into a closed form one. Recently, an extended tanh-function method and symbolic computation have been suggested for solving the new coupled modified KDV equations to obtain four kinds of soliton solutions [18]. This method has some merits in contrast with the tanh-function method. It only uses a simpler algorithm to yield an algebraic system, also it yields singular soliton solutions with no extra effort [19–22]. ADM has been previously implemented to obtain exact solutions of this system [23]. Then, variational iterations method proposed by He [24] was used to solve different types such as one dimensional Burgers’ equation and coupled Burgers’ equations [17]. As we will show, ADM, even MADM, have weakness that give approximate solution with acceptable accuracy, only in limited interval and outside it, high errors are occurred. To overcome this weakness, some authors have used Pade approximants with ADM in [25]. The first connection between ADM and Pade approximants was established in [26–29]. In this paper, we use the Pade approximants with ADM and MADM to improve these methods and to obtain approximate solutions with high accuracy in larger interval both for homogeneous and inhomogeneous two dimensional heat equations. Also, we use this technique to obtain such approximate solutions for coupled Burgers’ equations. 2. Adomian decomposition method 2.1. ADM applying to parabolic equations We begin with the two dimensional heat equation in the operator form Lt u  ru ¼ gðx; y; tÞ;

ð2:1Þ

where RLt u ¼ o=ot; r ¼ o2 =ox2 þ o2 =oy 2 and g is a given function. Assume that the inverse operator t 1 L1 on both sides of Eq. (2.1) yields, t ¼ 0 ðÞ dt exists and easily obtained. Applying the inverse operator Lt 1 uðx; y; tÞ ¼ uðx; y; 0Þ þ L1 t ðruÞ þ Lt gðx; y; tÞ:

ð2:2Þ

The ADM assumes that the unknown function u(x, y, t) can be expressed by an infinite series of the form [4] 1 X uðx; y; tÞ ¼ uk : ð2:3Þ k¼0

In the case of homogeneous two dimensional heat equation which implies that gðx; y; tÞ ¼ 0, the individual terms of u are obtained using the recursive algorithm u0 ¼ uðx; y; 0Þ; ukþ1 ¼ L1 t ðruk Þ;

ð2:4Þ

k P 0:

In the case of inhomogeneous two dimensional heat equation with nonzero function g, we use the modified Adomian decomposition method (MADM) [30,31]. In this new modification, gðx; y; tÞ is replaced by a series of infinite components. Wazwaz [32,33] suggests that g can be expressed in Taylor series with respect to the variable t as 1 X g¼ gk : ð2:5Þ k¼0

So, a new recursive relationship is expressed in the form u0 ¼ g 0 ; ukþ1 ¼ gkþ1 þ L1 t ðruk Þ;

k P 0:

ð2:6Þ

1036

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

In other words, the components un ðxÞ; n P 0 can be obtained by using u0 ¼ g 0 ; u1 ¼ g1 þ L1 t ðru0 Þ; u2 ¼ g2 þ L1 t ðru1 Þ; .. . unþ1 ¼ gnþ1 þ L1 t ðrun Þ; .. .

ð2:7Þ

Convergence aspects of the ADM have been investigated by Adomian and Cherruault [34]. For later numerical computation, let the expression n X un ðx; y; tÞ ¼ uk ; ð2:8Þ k¼0

denotes the n-term approximation to u(x, y, t). 2.2. ADM applying to coupled Burgers’ equations We will consider the system of Burgers’ equations in the operator form  Lt u  Lxx u  2uLx u þ Lx uv ¼ 0; Lt v  Lxx v  2vLx v þ Lx uv ¼ 0;

ð2:9Þ

where Lx ¼ o=ox; Lxx ¼ o2 =ox2 and Lt ¼ o=ot. The solutions of which are to be obtained subject to the initial conditions uðx; 0Þ ¼ sin x;

vðx; 0Þ ¼ sin x:

ð2:10Þ

Clearly, the exact solutions of this system are uðx; tÞ ¼ expðtÞ sinðxÞ;

vðx; tÞ ¼ expðtÞ sinðxÞ:

ð2:11Þ

L1 t

which is simply onefold integration operator. In the case, defined by We define R t the inverse operator L1 ¼ ðÞ dt we apply the inverse operator to both sides of the system (2.9) to get: t 0 1 1 uðx; yÞ ¼ uðx; 0Þ þ L1 t Lxx u þ Lt ð2M 1 ðuÞÞ  Lt Lx N ðu; vÞ; 1 1 vðx; yÞ ¼ vðx; 0Þ þ L1 t Lxx v þ Lt ð2M 2 ðvÞÞ  Lt Lx N ðu; vÞ;

ð2:12Þ

where M 1 ðuÞ ¼ uux , M 2 ðvÞ ¼ vLx v, N ðu; vÞ ¼ uv are the nonlinear terms. The ADM [4] consists of representing u(x, t) and v(x, t) in the decomposition forms given by 1 X un ; u¼ n¼0

v¼

1 X

ð2:13Þ vn ;

n¼0

respectively. The components un and vn ; n P 0 can be determined easily in a recursive manner. The nonlinear operators M 1 ; M 2 and N, can be defined by the infinite series of Adomian’s polynomials [4,6,35,36] 1 X An ; M 1 ðuÞ ¼ n¼0

M 2 ðuÞ ¼

1 X

N ðu; vÞ ¼

Bn ;

n¼0 1 X n¼0

Cn:

ð2:14Þ

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

1037

The Adomian polynomials can be generated for all forms of nonlinearity. The Adomian polynomials An, Bn and Cn are generated according to the following: " , !# 1 X n n k An ðu0 ; u1 ; . . . ; un Þ ¼ ð1=n!Þ d dk M 1 k uk ; " , Bn ðv0 ; v1 ; . . . ; vn Þ ¼ ð1=n!Þ dn

dkn M 2

k¼0 1 X

!# k¼0 kk v k

" , n

C n ðu0 ; u1 ; . . . ; un ; v0 ; v1 ; . . . ; vn Þ ¼ ð1=n!Þ d

ð2:15Þ

;

k¼0

n

dk N

k¼0 1 X k¼0

k

k uk ;

1 X k¼0

!# k

k vk

: k¼0

For these formulae it is easy to set a computer code to get as many polynomial as we need in the calculation. We can give Adomian’s polynomials An ; Bn ; C n n X An ¼ uk o=oxðunk Þ; k¼0

Bn ¼ Cn ¼

n X k¼0 n X

vk o=oxðvnk Þ;

ð2:16Þ

uk vnk ;

k¼0

using the recursive formula: u0 ¼ uðx; 0Þ; v0 ¼ vðx; 0Þ; 1 1 unþ1 ¼ L1 t Lxx un þ Lt ð2An Þ  Lt Lx C n ;

ð2:17Þ

1 1 vnþ1 ¼ L1 t Lxx vn þ Lt ð2Bn Þ  Lt Lx C n ;

where n P 0. For numerical purposes, we use n X /n ðx; tÞ ¼ uk ; n P 0;

ð2:18Þ

k¼0

and wn ðx; tÞ ¼

n X

vk ;

n P 0:

ð2:19Þ

k¼0

3. Pade approximation The procedure P is ton seek a rational function for the series. Given a function f(z) expanded in a Maclaurin series f ðzÞ ¼ 1 n¼0 cn z , we can use the coefficients of the series to represent the function by a ratio of two polynomials A½L=M ðzÞ a0 þ a1 z þ    þ aL zL ¼ ; ½L=M ðzÞ b0 þ b1 z þ    þ bM zM B symbolized by [L/M] and called the Pade approximant. The basic idea is to match the series coefficients as far as possible. Even though the series has a finite region of convergence, we can obtain the limit of the function as x ! 1 if L ¼ M. We note that there are L þ 1 independent coefficients in the numerator and M+1 coefficients in the denominator. To make the system determinable, let b0 ¼ 1. We then have M independent coefficients in the denominator and L þ M þ 1 independent coefficients in all. Now the [L/M] approximant can fit the power series through orders 1; z; z2 ; . . . ; zLþM with an error of OðzLþMþ1 Þ. Consequently,

1038

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

ða0 þ a1 z þ    þ aL zL Þ ¼ ðb0 þ b1 z þ    þ bM zM Þðc0 þ c1 z þ   Þ: Equating coefficients of zLþ1 ; zLþ2 ; . . . ; zLþM in turn, we can write bM cLMþ1 þ bM1 cLMþ2 þ    þ b0 cLþ1 ¼ 0; bM cLMþ2 þ bM1 cLMþ3 þ    þ b0 cLþ2 ¼ 0; .. . bM cL þ bM1 cLþ1 þ    þ b0 cLþM ¼ 0: Setting b0 ¼ 1, we have M linear equations for the M coefficients in the denominator 0

cLMþ1

cLMþ2

...

Bc B LMþ2 B B @

cLMþ3 .. .

...

cL

cLþ1

cL

10

bM

1

0

cLþ1

1

CB b C Bc C CB M1 C B Lþ2 C CB . C ¼  B . C : CB . C B . C A@ . A @ . A b1 cLþM . . . cLþM1 cLþ1 .. .

We invert the matrix on the left and solve the resulted system for the bi for i ¼ 1; . . . ; M. Since we know the c0 ; c1 ; . . ., we can equate coefficients of 1; z; z2 ; . . . ; zL to get a0 ; a1 ; . . . ; aL . Thus we have a0 ¼ c 0 ; a1 ¼ c 1 þ b1 c 0 ; a2 ¼ c 2 þ b1 c 1 þ b2 c 0 ; .. . aL ¼ c L þ

minfM;Lg X

bi cLi :

i¼0

Thus the numerator and denominator f ðzÞ ¼ c0 þ c1 z þ c2 z2 þ   , we have 1z ½1=1 ¼ ba00 þa ; þb1 z

of

the

Pade

approximants

are

determined.

For

lim ½1=1 ¼ ab11 ;

z!1 2

1 zþa2 z ; ½2=2 ¼ ab00 þa þb1 zþb2 z2 2

lim ½2=2 ¼ ab22 ;

z!1 3

1 zþa2 z þa3 z ; ½3=3 ¼ ab00 þa þb1 zþb2 z2 þb3 z3

lim ½3=3 ¼ ab33 ;

z!1

.. . lim ½m=m ¼ abmm :

z!1

A collection of Pade approximants formed by using a suitable set of values of L and M often provides a means of obtaining information about the function outside its circle of convergence, and of more rapidly evaluating the function within its circle of convergence. Every power series has a circle of convergence jzj ¼ R. If the given power series converges to the same function for jzj < R with 0 < R < 1, then a sequence of Pade approximants may converge for z 2 D where D is a domain larger than jzj < R. For a rigorous justification of this technique, see [35, Chapters 5–6]. Now we give an example: Example sffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 12 z 3 39 f ðzÞ ¼ ¼ 1  z þ z2     4 32 1 þ 2z

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

1039

The radius of convergence of this series is 12. The Pade approximant [1/1] is 1 þ 78 z ½1=1 ¼ : 1 þ 138 z Fig. 1 shows remarkable accuracy of Pade approximant [1/1] (using just three terms of the series) with respect to truncated Maclaurin series. Rapid convergence of Pade approximants at high order, may be proved using the Stieltjes series methods introduced in [35, Chapter 5]. Because of the important role of exponential function as the solution of the most differential equations, considerable attention has been paid to this function. Also, explicit forms for its Pade approximants [35, Chapter 1] lead to better investigation of theoretical aspects of convergence of Pade approximants to this function. Theorem 1 [35, Chapter 1]. The determinant of the matrix 0 1 cLMþ1 cLMþ2 . . . cL Bc cLþ1 C B LMþ2 cLMþ3 . . . C B .. C .. B C @ . A . cL

cLþ1

...

cLþM1

related to exponential function is denoted by C(L/M) and is evaluated as CðL=MÞ ¼ ð1Þ

MðM1Þ=2

M Y k¼1

1 : k!ðk þ 1Þ!    ðk þ L  1Þ!

Notice that the sign of C(L/M) is (-1)M(M1)/2, which does not depend on L; for M ¼ 1; 2; 3; . . ., the signs are þ; ; ; þ; þ; ; ; . . ., and this pattern of signs of C(L/M) characterizes a class of functions known as Po´lya Q1 1 frequency series [37]. Every function of the class has the representation f ðzÞ ¼ a0 ecz j¼1 ð1 þ aj zÞð1  bj zÞ with a0 > 0; c P 0; aj P 0; bj P 0, and 1 X ðaj þ bj Þ is convergent: j¼1

The present point of interest is that these functions have not only convergent Pade approximants, but also the numerators and denominators of the Pade approximants separately converges as k ! 1.

Fig. 1. Comparison of f(z) with Pade approximant and Truncated Maclaurin series.

1040

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

Theorem 2 [38]. If f(z) is a Po´lya frequency series defined by . . ., and Lk ! 1, M k ! 1 with (M k =Lk Þ ! x as k ! 1, then  Q 1 c A½Lk =M k  ðzÞ ! a0 exp 1þx z ð1 þ aj zÞ; B½Lk =M k  ðzÞ ! a0 exp



j¼1

cx z 1þx

Q 1

ð1  bj zÞ;

j¼1

uniformly on any compact region of the z-plane. Corollary. If f ðzÞ ¼ ecz ,   cz ½Lk =M k  A ! exp 1þx

and

B

½Lk =M k 



 cxz ! exp : 1þx

As seen from Theorem 2 and the Corollary, two points are considerable. Firstly, the convergence is uniform in any compact region in the case of Pade approximation, while truncated Maclaurin series is valid only in a near neighborhood of zero point, also convergent is in general pointwise. These mean that in the case of Pade approximation we have larger interval of convergence. Secondly, higher order of Pade approximants, give better approximations to the exponential function in the sense of infinity norm. Accordingly, the main advantage in the approach of using ADM–PADE technique comes from the above Theorem. 4. Applications 4.1. The inhomogeneous two dimensional heat equation Consider the following equation: Lt u ¼ ru  sin x sin y et  4;

ð4:1Þ

with initial condition uðx; y; 0Þ ¼ sin x sin y þ x2 þ y 2 ;

0 < x; y < 1:

ð4:2Þ

The exact solution is clearly uðx; y; tÞ ¼ sin x sin yet þ x2 þ y 2 :

ð4:3Þ

Using MADM, we obtain the resulting components as below: u0 ¼ sin x sin y þ x2 þ y 2 ; u1 ¼ t sin x sin y; .. .

ð4:4Þ

and so on. Now, we apply MADM–PADE technique to approximate u10 ðx; y; tÞ using the rational approximation r[5/ 5] which takes the form: 30240ðsin x sin y þ x2 þ y 2 Þ þ 15120tð sin x sin y þ x2 þ y 2 Þ r½5=5ðx; y; tÞ ¼ 30240 þ 15120t þ 3360t2 þ 420t3 þ 30t4 þ t5 3360t2 ðsin x sin y þ x2 þ y 2 Þ þ 420t3 ð sin x sin y þ x2 þ y 2 Þ þ 30240 þ 15120t þ 3360t2 þ 420t3 þ 30t4 þ t5 4 30t ðsin x sin y þ x2 þ y 2 þ t5 ð sin x sin y þ x2 þ y 2 ÞÞ þ : ð4:5Þ 30240 þ 15120t þ 3360t2 þ 420t3 þ 30t4 þ t5 The difference between exact solution u(x, y, t) and u10 ðx; y; tÞ at points t ¼ 2; 3; 4; 5, also difference between u(x, y, t) and MADM–PADE approximation r[5/5] at same points, are illustrated in Figs. 2 and 3. These

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

1041

Fig. 2. Error function uðx; y; tÞ  u10 ðx; y; tÞ for t ¼ 2; 3; 4; 5.

figures show when the interval of convergence extends by increasing variable t from 2 to 5, the accuracy of MADM decreases considerably. On the other hand, as shown in those figures, the high accuracy of solution given by MADM–PADE technique remains remarkable. 4.2. The homogeneous two dimensional heat equation Let us consider the equation Lt u ¼ ru

ð4:6Þ

subject to the initial condition uðx; y; 0Þ ¼ sin x sin y;

0 < x; y < p:

ð4:7Þ

The exact solution is uðx; y; tÞ ¼ e2t sin x sin y:

ð4:8Þ

In order to solve this equation using ADM, we simply use (2.4) to find the zeroth component of u0 as u0 ¼ sin x sin y;

ð4:9Þ

and the remaining components u1 ; u2 ; . . . are computed by a recursive scheme. The well known software Maple 9 is used to compute the components. By using (2.4) some of the symbolically computed components are as follow:

1042

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

Fig. 3. Error function uðx; y; tÞ  r½5=5ðx; y; tÞ for t ¼ 2; 3; 4; 5.

u1 ¼ 2t sin x sin y; u2 ¼ 2t2 sin x sin y; .. .

ð4:10Þ

Thus, other components can be computed easily. Truncated series solution u10 ðx; y; tÞ and ADM–PADE approximation r[5/5] take the forms: u10 ðx; y; tÞ ¼ sin x sin y  2t sin x sin y þ 2t2 sin x sin y  4=3t3 sin x sin y þ 2=3t4 sin x sin y  4=15t5 sin x sin y þ 4=45t6 sin x sin y  8=315t7 sin x sin y þ 2=315t8  4=2835t9 sin x sin y þ 4=14175t10 sin x sin y; r½5=5ðx; y; tÞ ¼

945 sin x sin y  945t sin x sin y þ 420t2 sin x sin y  105t3 sin x sin y þ 15t4 sin x sin y 945 þ 945t þ 420t2 þ 105t3 þ 15t4 þ t5 5 t sin x sin y  ; 945 þ 945t þ 420t2 þ 105t3 þ 15t4 þ t5

ð4:11Þ

ð4:12Þ

respectively. The related results are illustrated in Figs. 4 and 5. This evidently shows the noticeable advantages of ADM–PADE technique over the classic ADM [39–43].

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

1043

Fig. 4. Error function uðx; y; tÞ  u10 ðx; y; tÞ for t ¼ 2; 3; 4; 5.

4.3. The coupled Burgers’ equations According to (2.17) we can evaluate the approximate solutions /n(x,t) and wn(x,t), by using the n-term approximations, where the components are produced as u0 ¼ sin x; u1 ¼ t sin x; .. . v0 ¼ sin x;

ð4:13Þ

v1 ¼ t sin x; .. . and other components can be generated with the aid of Maple package. Because of equality of truncated series solution /n(x, t) and wn(x, t) produced by ADM, also equality of exact solutions u(x, t) and v(x, t), the ADM– PADE technique is considered only for solution u(x, t). A comparison between ADM truncated series solution

1044

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

Fig. 5. Error function uðx; y; tÞ  r½5=5ðx; y; tÞ for t ¼ 2; 3; 4; 5.

/10 ðx; tÞ and exact solution u(x, t) at x ¼ 1, with the aid of Fig. 6, shows that ADM gives good approximation in small interval of convergence and high errors occur beyond the small interval. ADM–PADE approximations r[3/3] and r[5/5], take the forms:

Fig. 6. Error function uðx; tÞ  /10 ðx; tÞ at x ¼ 1 for t ¼ 3; 4; 5.

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

120 sin x  60t sin x þ 12t2 sin x  t3 sin x ; 120 þ 60t þ 12t2 þ t3 30240 sin x  15120t sin x þ 3360t2 sin x  420t3 sin x þ 30t4 sin x  t5 sin x : r½5=5ðx; tÞ ¼ 30240 þ 15120t þ 3360t2 þ 420t3 þ 30t4 þ t5

r½3=3ðx; tÞ ¼

1045

ð4:14Þ ð4:15Þ

Figs. 7 and 8 illustrate how much the results are improved at x ¼ 1 than using ADM. Also it is noticeable that increasing the order of pade approximants, leads to better accuracy and larger interval of convergence. The

Fig. 7. Error function uðx; tÞ  r½5=5ðx; tÞ at x ¼ 1 for t ¼ 3; 4; 5.

Fig. 8. Error function uðx; tÞ  r½3=3ðx; tÞ at x ¼ 1 for t ¼ 3; 4; 5.

Fig. 9. Error function uðx; tÞ  r½5=5ðx; tÞ.

1046

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

Fig. 10. Error function uðx; tÞ  /10 ðx; tÞ.

difference between exact solution u(x, t) and ADM–PADE approximant r[5/5], also difference between u(x, t) and /10 ðx; tÞ, are illustrated in Figs. 9 and 10, respectively. Again, these figures show the advantage of ADM– PADE technique over the ADM. 5. Conclusion In this work, we employed the ADM–PADE and MADM–PADE techniques, for solving approximately homogeneous and inhomogeneous two dimensional heat equations. Also we implemented ADM–PADE technique to find approximate solution of coupled Burgers’ equations. The results show that these techniques, increase efficiently the accuracy of approximate solution and lead to convergence with a rate faster than using ADM and MADM. Employing higher order pade approximations produces more efficient results. In this work, we used well known software Maple to calculate the series and the rational functions obtained from the proposed techniques. References [1] K. Abbaoui, Y. Cherruault, Convergence of Adomians method applied to differential equations, Comput. Math. Appl. 102 (1999) 77– 86. [2] M. Dehghan, Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications, Appl. Math. Comput. 157 (2) (2004) 549–560. [3] M. Dehghan, The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications, Int. J. Comput. Math. 81 (2004) 25–34. [4] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, 1994. [5] J.M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. I (1948) 171–199. [6] E. Babolian, Sh. Javadi, Restarted Adomian method for algebraic equations, Appl. Math. Comput. 146 (2–3) (2003) 533–541. [7] J.D. Cole, On a quasilinear parabolic equations occurring in aerodynamics, Quart. Appl. Math. 9 (1951) 225–236. [8] S.E. Esipov, Coupled Burgers’ equations: a model of polydispersive sedimentation, Phys. Rev. E 52 (1995) 3711–3718. [9] J. Nee, J. Duan, Limit set of trajectories of the coupled viscous Burgers equations, Appl. Math. Lett. 11 (1) (1998) 57–61. [10] J.D. Fletcher, Generating exact solutions of the two-dimensional Burgers equations, Int. J. Numer. Meth. Fluids 3 (1983) 213–216. [11] P.C. Jain, D.N. Holla, Numerical solution of coupled Burgers’ equations, Int. J. Numer. Meth. Eng. 12 (1978) 213–222. [12] F.W. Wubs, E.D. de Goede, An explicit–implicit method for a class of time-dependent partial differential equations, Appl. Numer. Math. 9 (1992) 157–181. [13] A.R. Bahadir, A fully implicit finite-difference scheme for two-dimensional Burgers’ equations, Appl. Math. Comput. 137 (2003) 131– 137. [14] A.A. Soliman, New numerical technique for Burgers’ equation based on similarity reductions, in: International Conference on Computational Fluid Dynamics, Beijing, China, October, vol. 1720, 2000, pp. 559–566. [15] C.J. Fletcher, A comparison of finite element and finite difference solutions of the one and two dimensional Burgers’ equations, J. Comput. Phys. 51 (1983) 159–188.

M. Dehghan et al. / Applied Mathematics and Computation 189 (2007) 1034–1047

1047

[16] S.F. Radwan, Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers equation, J. Comput. Appl. Math. 174 (2004) 383–397. [17] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger’s and coupled Burgers equations, J. Comput. Appl. Math. 181 (2) (2005) 245–251. [18] E. Fan, Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A 282 (2001) 18–22. [19] Y.T. Wu, X.G. Geng, X.B. Hu, S.M. Zhu, A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations, Phys. Lett. A 255 (1999) 259–264. [20] R. Hirota, J. Satsuma, Soliton solutions of a coupled Korteweg–de Vries equation, Phys. Lett. A 85 (1981) 407–408. [21] J. Satsuma, R. Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy, J. Phys. Soc. Jap. 51 (1982) 3390–3397. [22] E.G. Fan, H.Q. Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998) 403–406. [23] D. Kaya, An explicit solution of coupled viscous Burgers’ equation by decomposition method, Int. J. Math. Math. Sci. 27 (2001) 675– 680. [24] J.H. He, A new approach to nonlinear partial differential equations, Comm. Nonlinear Sci. Numer. Simul. 2 (1997) 230–235. [25] Tamer A. Abassy, Magdy A. El-Tawil, Hassan K. Saleh, The solution of Burgers’ and good Boussinesq using ADM–Pade’ technique, Chaos, Solitons & Fractals, in press. [26] H.N.A. Ismail, K.R. Raslan, G.S.E. Salem, A.A. Abd Rabb, Comparison study between restrictive Taylor, restrictive Pade’ approximations and Adomian decomposition method for the solarity wave solution of the general KdV equation, Appl. Math. Comput. 167 (2005) 849–869. [27] Y.C. Jiao, Y. Yamamoto, C. Dang, Y. Hao, An after treatment technique for improving the accuracy of Adomian decomposition method, Comput. Math. Appl. 43 (2002) 783–798. [28] A.M. Wazwaz, Analytical approximations and Pade’ approximants for Volterra’s population model, Appl. Math. Comput. 100 (1999) 13–25. [29] A.M. Wazwaz, The modified decomposition method and Pade’ approximants for solving the Thomas–Fermi equation, Appl. Math. Comput. 102 (1999) 11–19. [30] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988) 501–544. [31] G. Adomian, Modification of the decomposition approach to heat equation, J. Math. Anal. Appl. 124 (1987) 290–291. [32] A.M. Wazwaz, S.M. El-Sayed, A new modification of Adomian decomposition method for linear and nonlinear operators, Appl. Math. Comput. 122 (2001) 393–405. [33] A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1999) 77–86. [34] Y. Cherruault, G. Adomian, Decomposition method: a new proof of convergence, Math. Comp. Model. 18 (1993) 103–106. [35] G. Adomian, R. Rach, Modified Adomian polynomials, Math. Comput. Model. 24 (11) (1996) 39–46. [36] G.A. Baker Jr., P. Graves-morris, Pade Approximants, Cambridge, 1997. [37] I.J. Schoenberg, On the Po´lya frequency functions I: The totally positive functions and their Laplace transforms, J. d’ Anal. Math. 1 (1951) 331–374. [38] R.J. Arms, A. Edrei, Pade table and continued fractions generated by totally positive sequences, in: Mathematical Essayes, Ohio University Press, 1970, pp. 1–21. [39] M. Dehghan, M. Tatari, The use of Adomian decomposition method for solving problems in calculus of variations, Math. Probl. Eng. (2006) 1–12. [40] M. Dehghan, B. Hashemi, Solution of the fully fuzzy linear systems using the decomposition procedure, Appl. Math. Comput. 182 (2006) 1568–1580. [41] M. Tatari, M. Dehghan, Numerical solution of Laplace in a disk using the Adomian decomposition method, Phys. Scripta 72 (2005) 345–348. [42] M. Dehghan, The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure, Int. J. Comput. Math. 81 (2004) 979–989. [43] M. Tatari, M. Dehghan, M. Razzaghi, Application of the Adomian decomposition method for the Fokker-Planck equation, Math. Comput. Model. 45 (2007) 639–650.