Applied Mathematics and Computation 166 (2005) 426–433

www.elsevier.com/locate/amc

An implementation of the ADMfor generalized one-dimensional

Klein-Gordon equation

Dogan Kaya

Department of Mathematics, Firat University, Elazig 23119, Turkey

Abstract

In this study, a decomposition method for approximating the solution of the gener-

alized one-dimensional Klein-Gordon equation is implemented. To illustrate the appli-

cation of this method, numerical results are derived by using the calculated components

of the decomposition series. The obtained results are found to be in good agreement

with the exact solution.

� 2004 Elsevier Inc. All rights reserved.

Keywords: The decomposition method; The generalized one-dimensional Klein-Gordon equation;

Traveling wave solution; Solitary wave solution

1. Introduction

The generalized one-dimensional Klein-Gordon (godKG) equation

utt � kuxx þ b1uþ b2uþ1 þ b3u2pþ1 ¼ 0 ð1Þ

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

doi:10.1016/j.amc.2004.06.103

E-mail addresses: dkaya@firat.edu.tr, dkaya36@yahoo.com

D. Kaya / Appl. Math. Comput. 166 (2005) 426–433 427

is given in [1]. In the work of Zhang et al. [1], the author obtained exact trave-

ling wave solutions to the godKG equation (1). Eq. (1) represent a nonlinear

model of longitudinal wave propagation of elastic rods when p = 1 [2]. In the

work of Refs. [3–5], the authors obtained exact traveling wave solutions to

Eq. (1) for the values p = 2, 3, 5, respectively.

Finding explicit exact and numerical solutions of nonlinear equations effi-ciently is of major importance and has widespread applications in numerical

methods and applied mathematics. In this study, we will implement the Ado-

mian decomposition method (in short ADM) [6–8] to find exact solution and

approximate solutions to the godKG equation for a given nonlinearities.

Unlike classical techniques, the decomposition method leads to an analytical

approximate and exact solutions of the nonlinear equations easily and ele-

gantly without transforming the equation or linearization of the problem

and with high accuracy, minimal calculation, and avoidance of physically unre-alistic assumptions. The method has features in common with many other

methods, but it is distinctly different on close examination, and one should

not be mislead by apparent simplicity into superficial conclusions [7].

Our aim is in this implementation to show how the ADM is effective for

using this type of equation with any order nonlinear terms. In this paper, the

godKG equation (1) for the values p = 1, 2, 3, 5 can be handled more easily,

quickly, and elegantly by implementing the ADM rather than the traditional

methods for finding analytical as well as numerical solutions.

2. Analysis of the method

In this section we outline the steps to obtain analytic solution of godKG

equation (1) using the ADM. First, we write the godKG equation in the stand-

ard operator form

Ltu� kLxuþ b1uþ b2upþ1 þ b3u2pþ1 ¼ 0; ð2Þwhere the notations t ¼ o

ot2 and x ¼ oox2 symbolize the linear differential opera-

tors. The inverse operator �1t exists and it can conveniently be taken as the two-

fold integration operator �1t . Thus, applying the inverse operator �1

t to (2) yields

L�1t Ltu ¼ kL�1

t ðLxu� ðb1uþ b2upþ1 þ b3u2pþ1ÞÞ: ð3ÞTherefore, it follows that

uðx; tÞ ¼ uðx; 0Þ þ tutðx; 0Þ þ L�1t ðkLxu� ðb1uþ b2upþ1 þ b3u2pþ1ÞÞ: ð4Þ

Now we decompose the unknown function u(x, t) a sum of components defined

by the series

uðx; tÞ ¼X1n¼0

unðx; tÞ: ð5Þ

428 D. Kaya / Appl. Math. Comput. 166 (2005) 426–433

The zeroth component is usually taken to be all terms arise from the initial

conditions, i.e.,

u0 ¼ uðx; 0Þ þ tutðx; 0ÞÞ: ð6Þ

The remaining components un(x, t), nP1, can be completely determinedsuch that each term is computed by using the previous term. Since u0 is

known,

un ¼ L�1t ðkLxun�1 � ðb1un�1 þ b2An�1 þ b3Bn�1ÞÞ; n P 1; ð7Þ

where upþ1 ¼P1

n¼0Anðu0; u1; . . . ; unÞ and u2pþ1 ¼P1

n¼0Bnðu0; u1; . . . ; unÞ. The

components, An and Bn are called the Adomian polynomials, these polynomials

can be calculated for all forms of nonlinearity according to specific algorithms

constructed by Adomain [6,9]. For this specific nonlinearity, we use the general

formula for An (similarly Bn polynomials) polynomials as

An ¼1

n!dn

dknfX1k¼0

kkuk

!" #k¼0

; n P 0: ð8Þ

This formula make it easy to set computer code to get as many polynomial as

we need in the calculation of the numerical as well as analytical solutions. For

sake of the easy follow of the reader, we could choice the nonlinear terms ofEq. (1) as Nu = up+1 and Mu = u2p+1 and then we can construct few terms of

the Adomian polynomials by using (8) as following

A0 ¼ upþ10 ; A1 ¼ ðp þ 1Þu1up0; A2 ¼

p þ 1

2up0ð2u2u0 þ pu21Þ;

A3 ¼p þ 1

6up�20 ð6pu2u1u0 þ 6u20u3 � pu31 þ p2u31Þ; . . .

and

B0 ¼ u2pþ10 ; B1 ¼ ð2p þ 1Þu1u2p0 ; B2 ¼ ð2p þ 1Þu2p�1

0 ðu2u0 þ pu21Þ;

B3 ¼2p þ 1

3u2p�20 ð6pu2u1u0 þ 3u20u3 � pu31 þ p2u31Þ; . . .

A slight modification to the ADM was proposed by Wazwaz [8] that givessome flexibility in the choice of the zeroth component u0 to be any simple term

and modify the term u1 accordingly. Since the computations in (7) depends

heavily on u0 the whole computations to find the solution will be simplified

considerably. For example an alternative scheme to (7) might be

D. Kaya / Appl. Math. Comput. 166 (2005) 426–433 429

u0 ¼ 0; u1 ¼ uðx; 0Þ þ tutðx; 0Þ þ L�1t ðkLxu0 � ðb1u0 þ b2A0 þ b3B0ÞÞ;

un ¼ L�1t ðkLxun�1 � ðb1un�1 þ b2An�1 þ b3Bn�1ÞÞ; n P 2: ð9Þ

Numerical computations of the godKG equation have often been repeated

in the literature. However, to show the effectiveness of the proposed decompo-

sition method and to give a clear overview of the methodology some examplesof the godKG equation (1) will be discussed in the following section.

3. Applications of the godKG equation

In this section we will be concerned with the solitary wave solutions of the

godKG equation

utt � kuxx � ðb1uþ b2upþ1 þ b3u2pþ1Þ ¼ 0;

uðx; 0Þ ¼ ðS1ð1þ tanhðRxÞÞÞ1p; ð10Þ

where pP1, R ¼ p2

ffiffiffiffiffiffiffi�b1a2�k

q, S1 ¼ �b2ðpþ1Þ

2b3ðpþ2Þ and b1 ¼b22ðpþ1Þ

b3ðpþ1Þ2, b2 5 0, b3(a2 � k) < 0,

a, k are arbitrary constants. Existence and derivations of such solutions have

been discussed for particular values of the constants [1–5].

In the first example, we will consider Eq. (11) for the special case p = 1 asso-

ciated the initial conditions

uðx; 0Þ ¼ S1ð1þ tanhðRðx� atÞÞÞ;utðx; 0Þ ¼ �aRS1 sech

2ðRðx� atÞÞ: ð11Þ

To find the solution of the initial value problem (11) and (12) we apply the

scheme (10). The Adomian polynomials An are computed according to (8). Per-

forming the integration we obtain the following

u0 ¼ 0; u1 ¼ �ðaRS1t sech2ðRxÞÞ þ S1ð1þ tanhðRxÞÞ; ð12Þ

u2 ¼�aRS1t3

12ð�b1 � 8kR2 � b1 coshð2RxÞ þ 4kR2 coshð2RxÞÞ sech4ðRxÞ

� sech3ðRxÞ S1t2 coshðRxÞ4

þ S1t2 sinhðRxÞ4

� �

� ðb1 þ 2kR2 þ b1 coshð2RxÞ � 2kR2 coshð2RxÞ þ 2kR2 sinhð2RxÞÞ;

ð13Þ

430 D. Kaya / Appl. Math. Comput. 166 (2005) 426–433

u3 ¼�aRS1t5

9603b21 þ 24b1kR

2 þ 528k2R4 þ 4b21 coshð2RxÞ�

þ 16b1kR2 coshð2RxÞ � 416k2R4 coshð2RxÞ þ b21 coshð4RxÞ

�8b1kR2 coshð4RxÞ þ 16k2R4 coshð4RxÞ

�sechðRxÞ6

� sech3ðRxÞ �ab2RS21t

3 coshðRxÞ3

� ab2RS21t

3 sinhðRxÞ3

� �

� sech2ðRxÞ b2S21t

2 coshð2RxÞ2

þ b2S21t

2 sinhð2RxÞ2

� �

� sech5ðRxÞ S1t4 coshðRxÞ192

þ S1t4 sinhðRxÞ192

� �

� ½�3b21 � 4b1kR2 � 88k2R4 þ 8a2b2R2S1 � 4b21 coshð2RxÞ

þ 96k2R4 coshð2RxÞ þ 8a2b2R2S1 coshð2RxÞ � b21 coshð4RxÞ

þ 4b1kR2 coshð4RxÞ � 8k2R4 coshð4RxÞ � 8b1kR

2 sinhð2RxÞ

� 80k2R4 sinhð2RxÞ � 8a2b2R2S1 sinhð2RxÞ � 4b1kR2 sinhð4RxÞ

þ 8k2R4 sinhð4RxÞ�: ð14Þ

In this manner the components of the decomposition series (5) are obtained as

many terms as we like. We could use the calculated terms (12)–(14) in the

decomposition series (5) or (16) and this series is exact to the last term, as

one can verify, of the Taylor series of the exact closed form solution

uðx; tÞ ¼ S1ð1þ tanhðRðx� atÞÞÞ ð15Þ

R ¼ 12

ffiffiffiffiffiffiffi�b1a2�k

q, S1 ¼ �b2

3b3, b1 ¼

2b22

9b3, b2 5 0, b3 (a2 � k) < 0, a, k are arbitrary

constants.

4. Experimental results for the godKG equation

The convergence of the decomposition series have investigated by severalauthors. The theoretical treatment of convergence of the decomposition meth-

od has been considered in the literature [10–15]. They obtained some results

about the speed of convergence of this method. In recent work of Abbaoui

et al. [16] have proposed a new approach of convergence of the decomposition

series. The authors have given a new condition for obtaining convergence of

the decomposition series to the classical presentation of the ADM in [16]. In

D. Kaya / Appl. Math. Comput. 166 (2005) 426–433 431

this work, we demonstrate the how approximate solutions of the godKG equa-

tions are close to corresponding exact solutions.

We use the ADM to solve the godKG equation (1). For numerical compar-

isons purposes, we consider various godKG equations (i.e., p = 1, 2, 3, 5).

Based on the ADM, we constructed the solution u(x, t) as

limn!1

/n ¼ uðx; tÞ; where /nðx; tÞ ¼Xnk¼0

ukðx; tÞ; n P 0 ð16Þ

and the recurrence relation is given as in (9) with (8).

In order to verify numerically whether the proposed methodology lead to

higher accuracy, we can evaluate the numerical solutions using the n-term

approximation (16). Table 1 shows the difference of analytical solution and

numerical solution of the absolute error of the godKG equations with various

values of the x and t. It is to be note that five terms only were used in evaluatingthe approximate solutions. We achieved a very good approximation with the

Table 1

The numerical results for /n(x, t) in comparison with the analytical solution for u(x, t) when k = 1.8,

b2 = �0.01, b3 = 2.5, p = 0.1, for the solitary wave solution of the Eq. (1)

tijxi 0.1 0.2 0.3 0.4 0.5

ju � /nj when p = 1

0.1 1.30104E�18 2.10335E�17 1.06902E�16 3.37187E�16 8.23343E�16

0.2 1.30104E�18 2.08167E�17 1.06685E�16 3.37187E�16 8.23560E�16

0.3 1.30104E�18 2.10335E�17 1.06902E�16 3.37621E�16 8.24210E�16

0.4 1.30104E�18 2.10335E�17 1.06685E�16 3.37404E�16 8.24861E�16

0.5 1.08420E�18 2.10335E�17 1.06685E�16 3.37837E�16 8.24861E�16

ju � /nj when p = 2

0.1 1.08982E�09 4.35919E�09 9.80800E�09 1.74361E�08 2.72435E�08

0.2 1.09037E�09 4.36142E�09 9.81302E�09 1.74451E�08 2.72574E�08

0.3 1.09093E�09 4.36365E�09 9.81804E�09 1.74540E�08 2.72714E�08

0.4 1.09149E�09 4.36588E�09 9.82306E�09 1.74629E�08 2.72853E�08

0.5 1.09205E�09 4.36811E�09 9.82809E�09 1.74718E�08 2.72993E�08

ju � /nj when p = 3

0.1 3.74514E�09 1.49802E�08 3.37048E�08 5.99183E�08 9.36203E�08

0.2 3.74762E�09 1.49901E�08 3.37271E�08 5.99579E�08 9.36822E�08

0.3 3.75010E�09 1.50001E�08 3.37494E�08 5.99976E�08 9.37442E�08

0.4 3.75258E�09 1.50100E�08 3.37717E�08 6.00373E�08 9.38062E�08

0.5 3.75506E�09 1.50199E�08 3.37940E�08 6.00770E�08 9.38682E�08

ju � /nj when p = 5

0.1 1.37060E�08 5.48235E�08 1.23352E�07 2.19291E�07 3.42640E�07

0.2 1.37085E�08 5.48337E�08 1.23375E�07 2.19332E�07 3.42704E�07

0.3 1.37111E�08 5.48439E�08 1.23398E�07 2.19373E�07 3.42704E�07

0.4 1.37136E�08 5.48540E�08 1.23421E�07 2.19413E�07 3.42831E�07

0.5 1.37161E�08 5.48642E�08 1.23444E�07 2.19454E�07 3.42894E�07

432 D. Kaya / Appl. Math. Comput. 166 (2005) 426–433

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 add-

ing new terms of the decomposition series.

Numerical approximations show a high degree of accuracy and in most

cases /n, the n-term approximation is accurate for quite low values of n. The

solutions are very rapidly convergent by utilizing the ADM. The numerical re-sults we obtained justify the advantage of this methodology, even in the few

terms approximation is accurate. Furthermore, as the decomposition method

does not require discretization of the variables, i.e. time and space, it is not af-

fected by computation round off errors and one is not faced with necessity of

large computer memory and time.

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

algorithm provides highly accurate numerical solutions without spatial discret-

izations for nonlinear partial differential equations. It is also worth noting thatthe advantage of the decomposition methodology displays a fast convergence

of the solutions. The illustrations show the dependence of the rapid conver-

gence 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 u(x, 0), we

may increase the accuracy of the series solution by computing more terms

which is quite easy using one of the symbolic programming packages Mathem-

atica, Matlab, . . . etc.The solutions are very rapidly convergent by utilizing the ADM. The

numerical results we obtained justify the advantage of this methodology. Fur-

thermore, as the decomposition method does not require discretization of the

variables, i.e. time and space, it is not effected by computation round off errors

and necessity of large computer memory and time. Clearly, the series solution

methodology can be applied to various type of linear or nonlinear ordinary

differential equations [17,18] and partial differential equations [19–29] as well.

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