1 July 2002

Physics Letters A 299 (2002) 201–206

www.elsevier.com/locate/pla

An application for a generalized KdV equationby the decomposition method

Dogan Kayaa,∗, Mohammed Aassilab

a Department of Mathematics, Firat University, Elazig 23119, Turkeyb Institut de Recherche Mathematique Avancee, 7 rue Rene Descartes, 67084 Strasbourg cedex, France

Received 28 January 2002; received in revised form 29 April 2002; accepted 7 May 2002

Communicated by A.R. Bishop

Abstract

The explicit solutions to a generalized Korteweg–de Vries equation (KdV for short) with initial condition are calculated byusing the Adomian decomposition method. Using this approach we obtained for the numerical solutions of initial-value KdVequation. Numerical illustrations on well-known KdV equation with the rational and solitary wave solutions indicate that thedecomposition method is efficient and accurate. In addition, an illustration of the self canceling phenomena is also given. 2002Elsevier Science B.V. All rights reserved.

AMS classification: 35Q53; 35B45

Keywords: Adomian decomposition method; Generalized KdV equation; Solitary wave solution; Self-canceling noise terms

1. Introduction

In studying long waves, tides, solitary waves, andrelated phenomena, one is led to an equation of theform

(1)ut + (m + 1)(m + 2)umux + uxxx = f (x, t),

wheref (x, t) is a given function andm = 1,2, . . . ,with u, ux , uxx → 0 as |x| → ∞. This equationis referred to as a generalized Korteweg–de Vriesequation [1]. If m = 0, m = 1, andm = 2, Eq. (1)becomes linearized KdV, nonlinear KdV and modifiedKdV equations, respectively [1–5].

* Corresponding author.E-mail address: dkaya@firat.edu.tr (D. Kaya).

Nonlinear phenomena play a crucial role in appliedmathematics and physics. The nonlinear problemsare solved easily and elegantly, without linearizingthe problem, by using the Adomian’s decompositionmethod (ADM for short) [6–8]. The nonlinear KdVequation has been the focus of considerable studiesby [9–12]. Gardner [11], developed a variational andits Hamiltonian formulation to handle this problem. Inaddition, Gardner et al. [12], introduced some variousmethods for exact solution of the KdV equation.In most of the traditional methods, the problem isfirst introduced by a linear one-dimensional auxiliaryproblem then solve the problem. The main difficultyin using such method is that the obtained integralequation cannot always be solved in terms of thesimple analytic functions [13].

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)00652-7

202 D. Kaya, M. Aassila / Physics Letters A 299 (2002) 201–206

In this Letter, the generalized KdV problem (1),homogeneous or inhomogeneous, will be handledmore easily, quickly, and elegantly by implementingthe ADM rather than the traditional methods for theexact solutions as well as for numerical solutions,without suffering traditional difficulty.

2. Analysis of the method

In the preceding section we have discussed particu-lar devices of the generalized KdV equation. For pur-poses of illustration of the ADM, in this study we shallconsider Eq. (1) in its standard form

(2)∂u

∂t+ (m + 1)(m + 2)um ∂u

∂x+ ∂3u

∂x3 = f (x, t),

where m = 1,2, . . . , the solution of which is to beobtained subject to the initial conditionu(x,0) =g(x). In this section we outlined the method to obtainanalytic solution of Eq. (2) using the ADM. Let usconsider the standard form of KdV equation (2) in anoperator form

(3)Ltu + (m + 1)(m + 2)Nu + Lxu = f (x, t),

where the notationNu = umux symbolizes the non-

linear term, the notationLt = ∂∂t

andLx = ∂3

∂x3 sym-bolizes the linear differential operator. Assuming theinverse of the operatorL−1

t exists and it can conve-niently be taken as the definite integral with respect tot from 0 to t , i.e.,L−1

t = ∫ t

0(·) dt . Thus, applying the

inverse operatorL−1t to (3) yields

L−1t Ltu = L−1

t

(f (x, t)

)

(4)

− (m + 1)(m + 2)L−1t Nu − L−1

t Lx(u).

Therefore, it follows that

u(x, t) = u(x,0) + L−1t

(f (x, t)

)(5)− (m + 1)(m + 2)L−1

t Nu − L−1t Lx(u).

We assign the zeroth component by

(6)u0 = u(x,0) + L−1t

(f (x, t)

),

which is defined by all terms that arise from theinitial condition and from integrating the source term.We then decompose the unknown functionu(x, t) by

a sum of components defined by the decompositionseries

(7)u(x, t) =∞∑

n=0

un(x, t),

and the nonlinear termNu = umux , we expressed inthe form of An Adomian’s polynomials; thusNu =umux = ∑∞

n=0 An where these polynomials can becalculated for all forms of nonlinearity accordingto specific algorithms constructed in [6,14]. In thisspecific nonlinearity, we use the general form offormula forAn polynomials

An = 1

n!dn

dλn

[Φ

( ∞∑k=0

λkuk

)]λ=0

, n � 0.

This formulae is easy to set computer code to get asmany polynomial as we need in the calculation of thenumerical as well as explicit solutions. For the easy tofollow of the reader, we can give the first few Adomianpolynomials are

A0 = um0 (u0)x,

A1 = mum−10 u1(u0)x + um

0 (u1)x,

A2 = mum−10 u2(u0)x + (

m(m − 1)/2)um−2

0 u21(u1)x

+ mum−10 u1(u1)x + um

0 (u2)x,

(8)

A3 = mum−10 u3(u0)x + m(m − 1)um−2

0 u1u2(u0)x

+ (m(m − 1)/2

)um−2

0 u31(u0)x

+ mum−10 u2(u1)x + mum−1

0 u1(u2)x

+ um0 (u3)x,

and so on, the other polynomials can be constructedin a similar way. The remaining componentsun(x, t),n � 1, can be completely determined such that eachterm is computed by using the previous term. Sinceu0is known,

u1 = −(m + 1)(m + 2)L−1t (A0) − L−1

t Lx(u0),

u2 = −(m + 1)(m + 2)L−1t (A1) − L−1

t Lx(u1),

...

(9)

un = −(m + 1)(m + 2)L−1t (An−1) − L−1

t Lx(un−1).

As a result, the series solution is given by

D. Kaya, M. Aassila / Physics Letters A 299 (2002) 201–206 203

u(x, t) = u0 −∞∑

n=1

{(m + 1)(m + 2)L−1

t (An−1)

(10)+ L−1t Lx(un−1)

},

whereL−1t is the previously given integration opera-

tor andAn−1 are defined appropriate Adomian’s poly-nomials by (8). The solutionu(x, t) must satisfy therequirements imposed by the initial conditions. TheADM provides a reliable technique that requires lesswork if compared with the traditional techniques.

Recently, Wazwaz [15] proposed that the construc-tion of the zeroth component of the decomposition se-ries can be defined in a slightly different way. In [15],he assumed that if the zeroth component isu0 = g, thefunctiong is possible to be divided into two parts suchasg1 andg2, then one can formulate the recursive al-gorithm in a form of a modified recursive scheme asfollows:

u0 = g1,

(11)

u1 = g2 − (m + 1)(m + 2)L−1t (A0)

− L−1t Lx(u0),

un+1 = −(m + 1)(m + 2)L−1t (An)

(12)− L−1t Lx(un), n � 1.

This type of modification is giving more flexibility tothe ADM in order to solve complicate nonlinear dif-ferential equations. In many case the modified schemeavoids the unnecessary computations, especially incalculation of the Adomian polynomials. Furthermore,sometimes we do not need to evaluate the so-calledAdomian polynomials or if we need to evaluate thesepolynomials the computation will be reduced veryconsiderably by using the modified recursive scheme.For more details of this MDM, one can see Ref. [15].For illustration purposes we will consider both ho-mogeneous and inhomogeneous KdV equations in thefollowing section. We will show that how the ADM iscomputationally efficient. To give a clear overview ofthe methodology, the following examples will be dis-cussed.

3. Applications

Example 1 For purposes of illustration of the ADMfor solving a general form of the classical KdV

equation [1] (i.e., form = 1), we shall consider theequation with initial condition

(13)ut − 6uux + uxxx = 0, u(x,0) = 6

x2 .

Proceeding as before, we determine the individualterms of the series (7)

u0 = 6

x2 ,

(14)u1 = 6L−1t (u0u0x ) − L−1

t Lx(u0) = −288

x5 t,

(15)

u2 = L−1t (u1u0x + u0u1x ) − L1

t Lx(u1)

= 6048

x8t2,

u3 = L−1t (u2u0x + u1u1x + u0u2x ) − L1

t Lx(u2)

(16)= −103480

x11 t3,

and so on, in this manner four components of thedecomposition series (7) were obtained of whichu(x, t) was evaluated to have the following expansion:

u(x, t) = u0 + u1 + u2 + u3 + · · ·

(17)

= 6

x2 − 288

x5 t + 6048

x8 t2 − 103648

x11 t3 + · · · .

This gives the exact solution

(18)u(x, t) = 6x(x3 − 24t)

(x3 + 12t)2 ,

which can be verified through substitution.

Example 2 We consider, the second KdV equation[16] has the travelling-wave solution

ut + 6uux + uxxx = 0,

(19)u(x,0) = 1

2sech2

(1

2x

).

Again, to find the solution of this equation, wesimply take the equation in an operator form exactly inthe same manner as the form of Eq. (5) and using (6)we find

(20)u0 = 1

2sech2

(1

2x

),

(21)

u1 = −6L−1t (u0u0x ) − L−1

t Lx(u0)

= t

2sech2

(x

2

)tanh

(x

2

),

204 D. Kaya, M. Aassila / Physics Letters A 299 (2002) 201–206

(22)

u2 = −6L−1t (u1u0x + u0u1x ) − L1

t Lx(u1)

= t2

8

(−2+ cosh(x))sech4

(x

2

),

(23)

u3 = −6L−1t (u2u0x + u1u1x + u0u2x ) − L1

t Lx(u2)

= t3

48sech5

(x

2

)(−11 sinh

(x

2

)+ sinh

(3x

2

)),

and so on, in this manner the rest of components ofthe decomposition series were obtained. Substituting(20)–(23) into (7) gives the solutionu(x, t) in a seriesform and in a close form solution by

(24)u(x, t) = 1

2sech2

(1

2(x − t)

).

This result can be verified through substitution.

Example 3 As an example of the application of theself canceling phenomena [15,17,18], let us seek theanalytic solution of the inhomogeneous KdV equationwith the initial condition

ut − uux + uxxx = −ex − t2e2x,

(25)u(x,0) = 1.

To obtain the decomposition solution subject toinitial condition given, we first used (27) in an operatorform in the same manner as form (3) and then weused (11) and (12) with (8) to determine the individualterms of the decomposition series, we get immediately

(26)u0 = 1− tex,

(27)

u1 = − t3

3e2x + L−1

t (u0u0x ) − L−1t Lx(u0)

= t2ex

2− t3e2x

3− t2ex

2+ t3e2x

3,

(28)un+1 = 0, n � 1.

It is obvious that the “noise” terms appear betweenthe components ofu1, and these are all cancelled. Asseen from Eq. (27), the closed form of the solutioncan be found very easily by proper selection ofg1andg2. In the case of right choice of these functions,the modified technique accelerate the convergence ofthe decomposition series solution by computing justu0 andu1 terms of the series. The termu0 provides

the exact solution asu(t, x) = 1− tex and this can bejustified through substitution.

For numerical comparisons purposes, we considertwo KdV equations. The first one has the rationalsolution and the second one has the travelling-wavesolution. Based on the ADM, we constructed thesolutionu(x, t) as

limn→∞ φn = u(x, t), where

(29)φn(x, t) =n∑

k=0

uk(x, t), n � 0

and the recurrence relation is given as in (9). More-over, the decomposition series (29) solutions are gen-erally converged very rapidly in real physical prob-lems [6]. The convergence of the decomposition se-ries have investigated by several authors. The theo-retical treatment of convergence of the decompositionmethod has been considered in the literature [19–23].They obtained some results about the speed of con-vergence of this method providing us to solve linearand nonlinear functional equations. In recent work Ab-baoui et al. [24] have proposed a new approach ofconvergence of the decomposition series. The authorshave given a new condition for obtaining convergenceof the decomposition series to the classical presenta-tion of the ADM in [24]. In this Letter, we demonstratehow approximate solutions of the KdV equations areclose to exact solutions.

In order to verify numerically whether the proposedmethodology lead to higher accuracy, we can evaluatethe numerical solutions using then-term approxima-tion (29). Tables 1 and 2 show the difference of ana-lytical solution and numerical solution of the absoluteerror. We also demonstrate the numerical exact solu-tions of Eq. (13) in Fig. 1(a), the corresponding ap-proximate numerical solution in Fig. 1(b) and Eq. (19)in Fig. 2(a), the corresponding approximate numericalsolution in Fig. 2(b). It is to be noted that 10 termsonly were used in evaluating the approximate solu-tions. We achieved a very good approximation withthe actual solution of the equations by using 10 termsonly of the decomposition derived above. It is evidentthat the overall errors can be made smaller by addingnew terms of the decomposition series.

Numerical approximations show a high degree ofaccuracy and in most casesφn, then-term approxima-tion is accurate for quite low values ofn. The solutions

D. Kaya, M. Aassila / Physics Letters A 299 (2002) 201–206 205

Table 1

The numerical results forφn(x, t) in comparison with the analytical solutionu(x, t) = 6x(x3−24t)(x3+12t)2

, for the rational solution of Eq. (13)

ti /xi 10 20 30 40 50

10 1.34054E-10 2.46066E-07 1.92821E-05 4.17266E-04 4.47261E-0320 4.34052E-21 8.74881E-18 7.44969E-16 1.73682E-14 1.44434E-1930 3.01498E-27 6.14510E-24 5.29002E-22 1.24657E-20 1.68754E-1440 1.28126E-31 2.61870E-28 2.26052E-26 5.34138E-25 6.20563E-2450 5.20260E-35 1.06438E-31 9.19705E-30 2.17532E-28 2.52977E-27

Table 2The numerical results forφn(x, t) in comparison with the analytical solutionu(x, t) = 1

2 sech2( 12(x − t)), for the travelling-wave solution of

Eq. (19)

ti /xi 0.1 0.2 0.3 0.4 0.5

0.1 0.00000E+00 1.13631E-13 8.86624E-12 1.86847E-10 1.90520E-090.2 2.22045E-16 2.27152E-13 1.89001E-11 4.28983E-10 4.76894E-090.3 1.66533E-16 2.95097E-13 2.51345E-11 5.84712E-10 6.67059E-090.4 5.55112E-17 3.06810E-13 2.65745E-11 6.28414E-10 7.28925E-090.5 1.11022E-16 2.65621E-13 2.33807E-11 5.61615E-10 6.61636E-09

Fig. 1. The numerical results forφ10(x, t): (a) in comparison with the analytical solutionsu(x, t), (b) for the rational solutions with the initialcondition of Eq. (13).

are very rapidly convergent by utilizing the ADM. Thenumerical results we obtained justify the advantage ofthis methodology, even in the few terms approxima-tion is accurate.

4. Conclusions

In this Letter, the ADM was used for homogeneousand inhomogeneous nonlinear KdV equations with

initial conditions. A clear conclusion can be drawfrom the numerical results that the ADM algorithmprovides highly accurate numerical solutions withoutspatial discretizations for nonlinear partial differentialequations. It is also worth noting that the advantageof the decomposition methodology displays a fastconvergence of the solutions. The illustrations showthat the rapid convergence is depend on the characterand behavior of the solutions just as in a closed formsolutions. A fast convergence of the solution may

206 D. Kaya, M. Aassila / Physics Letters A 299 (2002) 201–206

Fig. 2. The numerical results forφ10(x, t): (a) in comparison with the analytical solutionsu(x, t), (b) for the solitary wave solution with theinitial condition of Eq. (19).

be achieved by observing the self-canceling “noise”terms and by proper selection of the initial terms, thedemonstration of this case is shown in Example 3.

Finally, we point out that, for given equations withinitial valuesu(x,0), the corresponding analytical andnumerical solutions are obtained according to the re-currence equation (9) with (8) using MATHEMATICA .

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