a

uation isth easilyalculatedtion.

e

e andent the

Physics Letters A 348 (2006) 244–250

www.elsevier.com/locate/pl

The exact and numerical solitary-wave solutionsfor generalized modified Boussinesq equation

Dogan Kaya

Department of Mathematics, Firat University, Elazig 23119, Turkey

Received 20 February 2004; received in revised form 25 August 2005; accepted 26 August 2005

Available online 6 September 2005

Communicated by A.R. Bishop

Abstract

In this study, a decomposition method for approximating the solution of the generalized modified Boussinesq eqimplemented. By using this scheme, explicit exact solution is calculated in the form of a convergent power series wicomputable components. To illustrate the application of this method, numerical results are derived by using the ccomponents of the decomposition series. The obtained results are found to be in good agreement with the exact solu 2005 Elsevier B.V. All rights reserved.

MSC: 35Q53; 35B45

Keywords: Decomposition method; Generalized modified Boussinesq equation; Traveling wave solution; Solitary wave solution

1. Introduction

The generalized modified Boussinesq (GMB) equation

(1)utt − δuxxtt − (f (u)

)xx

= 0,

represent a nonlinear model of longitudinal wave propagation of elastic rods[1,2]. In the work of Bogolubsky[1],the author obtained exact solitary wave solutions to Eq.(1) for f (u) = b1u + b2u

p+1 + b3u2p+1 for the values

p = 2,3,5, respectively,[3–5]. Zhang and Ma[5] and Li and Zhang[4] derived some explicit solitary wavsolutions of(1) using the method of solving algebraic equations for the casesf (u) = b1u + b2u

3 + b3u5 and

f (u) = b1u + b2u2 + b3u

3.Finding explicit exact and numerical solutions of nonlinear equations efficiently is of major importanc

has widespread applications in numerical methods and applied mathematics. In this study, we will implem

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

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2005.08.074

D. Kaya / Physics Letters A 348 (2006) 244–250 245

the

solutionsroblemmericalre it isory andcessitybut it iserficial

withnderical

atore

ed by

no-n

Adomian decomposition method (in short ADM)[6–8] to find exact solution and approximate solutions toGMB equation for a given nonlinearityf (u).

Unlike classical techniques, the decomposition method leads to an analytical approximate and exactof the nonlinear equations easily and elegantly without transforming the equation or linearization of the pand with high accuracy, minimal calculation, and avoidance of physically unrealistic assumptions. As nutool the method provide us with numerical solution without discretization of the given equation and therefonot effected by computation round off errors and one is not faced with necessity of large computer memtime—straightforward to write computer codes in any symbolic programming language—not faced with neof large computer memory and time. The method has features in common with many other methods,distinctly different on close examination, and one should not be mislead by apparent simplicity into supconclusions[7].

Our aim is in this implementation to show how the ADM is effective for using this type of equationany order nonlinear terms. In this Letter, various GMB equations[4,9] can be handled more easily, quickly, aelegantly by implementing the ADM rather than the traditional methods for finding analytical as well as numsolutions.

2. Analysis of the method

In this section we outline the steps to obtain analytic solution of GMB equation(1) using the ADM. First wewrite the GMB equation in the standard operator form

(2)Ltu − Lx(Ltu) − Lxf (u) = 0,

where the notations Łt = ∂2/∂t2 and Łx = ∂2/∂x2 symbolize the linear differential operators. The inverse operŁ−1

t exists and it can conveniently be taken as the twofold integration operator Ł−1t . Thus, applying the invers

operator Ł−1t to (2) yields

(3)L−1t Ltu = L−1

t

(Lx(Ltu) + Lxf (u)

).

Therefore, it follows that

(4)u(x, t) = u(x,0) + tut (x,0) + L−1t

(Lx(Ltu) + Lxf (u)

).

Now we decompose the unknown functionu(x, t) a sum of components defined by the series

(5)u(x, t) =∞∑

n=0

un(x, t).

The zeroth component is usually taken to be all terms arise from the initial conditions, i.e.,

(6)u0 = u(x,0) + tut (x,0).

The remaining componentsun(x, t), n � 1, can be completely determined such that each term is computusing the previous term. Sinceu0 is known,

(7)un = L−1t

(Lx(Ltun−1) + LxAn−1

), n � 1,

wheref (u) = ∑∞n=0 An(u0, u1, . . . , un). The componentsAn are called the Adomian polynomials, these poly

mials can be calculated for all forms of nonlinearity according to specific algorithms constructed by Adomai[6,7].For this specific nonlinearity, we use the general formula forAn polynomials as

(8)An = 1

n!

[dn

dλnf

( ∞∑λkuk

)], n � 0.

k=0 λ=0

246 D. Kaya / Physics Letters A 348 (2006) 244–250

n of the

w of thect

e

ple,

to showgy some

ce

This formula makes it easy to set computer code to get as many polynomial as we need in the calculationumerical as well as analytical solutions. The first few Adomian polynomials for the nonlinearityf (u)

A0 = f (1)(u0), A1 = u1f(1)(u0), A2 = u2f

(1)(u0) + 1

2!u21f

(2)(u0),

(9)A3 = u3f(1)(u0) + u2u1f

(2)(u0) + 1

3!u31f

(3)(u0), . . .

and so on, the rest of the polynomials can be constructed in a similar manner. For sake of the easy folloreader, we could choice the nonlinear terms of Eq.(1) asNu = up+1 andMu = u2p+1 and then we can construfew terms of the Adomian polynomials by using(8) as follows

A0 = up+10 , A1 = (p + 1)u1u

p

0 , A2 = p + 1

2u

p−10

(2u2u0 + pu2

1

),

A3 = p + 1

6u

p−20

(6pu2u1u0 + 6u2

0u3 − pu31 + p2u3

1

), . . .

and

A0 = u2p+10 , A1 = (2p + 1)u1u

2p

0 , A2 = (2p + 1)u2p−10

(u2u0 + pu2

1

),

A3 = 2p + 1

3u

2p−20

(6pu2u1u0 + 3u2

0u3 − pu31 + 2p2u3

1

), . . . .

A slight modification to the ADM was proposed by Wazwaz[8] that gives some flexibility in the choice of thzeroth componentu0 to be any simple term and modify the termu1 accordingly. Since the computations in(7)depends heavily onu0 the whole computations to find the solution will be simplified considerably. For examan alternative scheme to(7) might be

u0 = 0, u1 = u(x,0) + tut (x,0) + L−1t

(Lx(Ltu0) + LxA0

),

(10)un = L−1t

(Lx(Ltun−1) + LxAn−1

), n � 2.

Numerical computations of the GMB equation have often been repeated in the literature. However,the effectiveness of the proposed decomposition method and to give a clear overview of the methodoloexamples of the GMB equation(1) will be discussed in the following section.

3. Applications of the GMB equation

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

(11)utt − δuxxtt − (b1u + b2u

p+1 + b3u2p+1)

xx= 0, u(x,0) = S1

(1+ tanh(Rx)

)1/p,

wherep � 1, R = p

√−b2

1+α

α2δ, α =

√b1 − b2

2(p+1)

b3(p+2)2 , S1 = b2(p+1)2b3(p+2)

andb1, b2, b3 are arbitrary constans. Existen

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 casep = 1 associated the initial conditions

(12)u(x,0) = S1(1+ tanh

(R(x − αt)

)), ut (x,0) = −αRS1 sech2

(R(x − αt)

).

To find the solution of the initial value problem(11) and (12)we apply the scheme(10). The Adomian polyno-mialsAn are computed according to(8). Performing the integration we obtain the following

(13)u0 = 0, u1 = −αRS1t sech2(Rx) + S1(1+ tanh(Rx)

),

D. Kaya / Physics Letters A 348 (2006) 244–250 247

uldm,

atment oftsch

(14)u2 = αb1R3S1t

3

3

(2− cosh(2Rx)

)sech4(Rx) − b1R

2S1t2 sech2(Rx) tanh(Rx),

u3 = αb21R

5S1t5

60

(−33+ 26 cosh(2Rx) − cosh(4Rx))sech6(Rx)

+ R2S1t2 sech5(Rx)

2

(3b2S1 cosh(Rx) − b2S1 cosh(3Rx) + 22b1δR

2 sinh(Rx)

− b2S1 sinh(Rx) − 2b1δR2 sinh(3Rx) − b2S1 sinh(3Rx)

)+ R4S1t

4 sech6(Rx)

24

(−24α2b2S1 + 16α2b2S1 cosh(2Rx) + 10b21 sinh(2Rx) − b2

1 sinh(4Rx))

+ αR3S1t3 sech6(Rx)

6

(−66b1δR2 + 3b2S1 + 52b1δR

2 cosh(2Rx) + 2b2S1 cosh(2Rx)

(15)− 2b1δR2 cosh(4Rx) − b2S1 cosh(4Rx) + 10b2S1 sinh(2Rx) − b2S1 sinh(4Rx)

),

in this manner the components of the decomposition series(5) are obtained as many terms as we like. We couse the calculated terms(13)–(15)in the decomposition series(5) or (21) and this series is exact to the last teras one can verify, of the Taylor series of the exact closed form solution

(16)u(x, t) = S1(1+ tanh

(R(x − αt)

))which can be find in Ref.[9].

In the second example, we will consider the GMB equation(11)with the values ofp = 4 of the initial conditions

(17)u(x,0) = S1(1+ tanh(Rx)

)1/4, ut (x,0) = −αRS1 sech2(Rx)

4(1+ tanh(Rx))3/4.

Again, to find the solution of this equation, we substitute in the scheme(10)

u0 = 0, u1 = S1(1+ tanh(Rx)

)1/4 + t−αRS1 sech2(Rx)

4(1+ tanh(Rx))3/4

(18)+t∫

0

t∫0

[(b1u0 + b2A0 + b3A0)xx + (u0)xxtt

]dt dt,

...

(19)un =t∫

0

t∫0

[(b1un−1 + b2An−1 + b3An−1)xx + (un−1)xxtt

]dt dt, n � 2,

whereAn−1 andAn−1 the Adomian polynomials are constructed by using(8). Performing the calculations in(10)using MATHEMATICA and substituting into(5) gives the exact solution

(20)u(x, t) = S1(1+ tanh

(R(x − αt)

))1/4.

4. Experimental results for the GMB equation

The convergence of the decomposition series have investigated by several authors. The theoretical treconvergence of the decomposition method has been considered in the literature[10–16]. They obtained some resulabout the speed of convergence of this method. In recent work of Abbaoui et al.[15] have proposed a new approa

248 D. Kaya / Physics Letters A 348 (2006) 244–250

gence ofw

riousADM,

uate then

ximationevident

ricalccurate.space, itory and

curatenotingstrations

ust as in a

Table 1The numerical results whenb1 = 2.5, b2 = b3 = 0.1 andδ = −0.1 for the solution of Eq.(11) for initial conditions (p = 1,4,8)

(xi , ti ) (0.1,0.1) (0.2,0.2) (0.3,0.3) (0.4,0.4) (0.5,0.5)

p = 14.99776×10−6 0.0000202832 0.0000462231 0.0000830855 0.0001310374.83686×10−6 0.0000196598 0.0000448677 0.0000807633 0.000127554.66713×10−6 0.0000190005 0.0000434306 0.0000782946 0.0001238324.48893×10−6 0.0000183066 0.0000419146 0.0000756841 0.0001198914.30264×10−6 0.0000175796 0.0000403228 0.0000729368 0.000115734p = 40.0000217526 0.0000823911 0.000171427 0.000274593 0.0003754310.0000226838 0.0000872195 0.000184316 0.000300001 0.0004169420.0000230287 0.0000898246 0.000192691 0.000318594 0.0004501480.0000227973 0.0000901751 0.000196331 0.000329766 0.000473880.0000220308 0.0000883635 0.000195276 0.000333299 0.000487405p = 80.000108922 0.000408284 0.000806773 0.00115561 0.001270940.000105885 0.000416609 0.000865895 0.00131236 0.001552230.000093885 0.000388162 0.000848841 0.00136179 0.001734420.0000754451 0.000329938 0.000763189 0.00130237 0.001793650.0000538642 0.000253394 0.000627997 0.00115178 0.00172759

of convergence of the decomposition series. The authors have given a new condition for obtaining converthe decomposition series to the classical presentation of the ADM in[15]. In this work, we demonstrate hoapproximate solutions of the GMB equations are close to corresponding exact solutions.

We use the ADM to solve the GMB equation(11). For numerical comparisons purposes, we consider vaGMB equations (i.e.,p = 1,4,8). Both of these equations have the solitary-wave solutions. Based on thewe constructed the solutionu(x, t) as

(21)limn→∞φn = u(x, t), whereφn(x, t) =

n∑k=0

uk(x, t), n � 0

and the recurrence relation is given as in(10)with (8).In order to verify numerically whether the proposed methodology lead to higher accuracy, we can eval

numerical solutions using then-term approximation(21). Tables 1 and 2show the difference of analytical solutioand numerical solution of the absolute error of the GMB equations with various values of thex andt . It is to benote that 5 terms only were used in evaluating the approximate solutions. We achieved a very good approwith the actual solution of the equations by using 5 terms only of the decomposition derived above. It isthat the overall errors can be made smaller by adding new terms of the decomposition series.

Numerical approximations show a high degree of accuracy and in most casesφn, then-term approximation isaccurate for quite low values ofn. The solutions are very rapidly convergent by utilizing the ADM. The numeresults we obtained justify the advantage of this methodology, even in the few terms approximation is aFurthermore, as the decomposition method does not require discretization of the variables, i.e., time andis not affected by computation round off errors and one is not faced with necessity of large computer memtime.

A clear conclusion can be draw from the numerical results that the ADM algorithm provides highly acnumerical solutions without spatial discretizations for nonlinear partial differential equations. It is also worththat the advantage of the decomposition methodology displays a fast convergence of the solutions. The illushow the dependence of the rapid convergence depend on the character and behavior of the solutions jclosed form solutions.

D. Kaya / Physics Letters A 348 (2006) 244–250 249

heackages

ify theon of theomputernlinear

Table 2The numerical results whenb1 = 2.5, b2 = b3 = 0.1 andδ = −0.1 for the solution of Eq.(11) for initial conditions (p = 1,4,8)

(xi , ti ) (0.01,0.01) (0.02,0.02) (0.03,0.03) (0.04,0.04) (0.05,0.05)

p = 15.06637× 10−8 2.02971× 10−7 4.57391× 10−7 8.1438× 10−7 1.27439× 10−6

5.05107× 10−8 2.02361× 10−7 4.56023× 10−7 8.11956× 10−7 1.27061× 10−6

5.03568× 10−8 2.01748× 10−7 4.54646× 10−7 8.09517× 10−7 1.26681× 10−6

5.0202× 10−8 2.0113× 10−7 4.53261× 10−7 8.07063× 10−7 1.26299× 10−6

5.00462× 10−8 2.00509× 10−7 4.51868× 10−7 8.04594× 10−7 1.25915× 10−6

p = 42.13244× 10−7 8.49684× 10−7 1.904× 10−6 3.37034× 10−6 5.24238× 10−6

2.14685× 10−7 8.55564× 10−7 1.91748× 10−6 3.39476× 10−6 5.28122× 10−6

2.16071× 10−7 8.61223× 10−7 1.93047× 10−6 3.4183× 10−6 5.31869× 10−6

2.174× 10−7 8.66657× 10−7 1.94296× 10−6 3.44096× 10−6 5.35479× 10−6

2.18672× 10−7 8.71865× 10−7 1.95494× 10−6 3.46272× 10−6 5.3895× 10−6

p = 81.08596× 10−6 4.32831× 10−6 9.69875× 10−6 0.0000171624 0.00002667781.09177× 10−6 4.35366× 10−6 9.76051× 10−6 0.0000172805 0.00002687521.09656× 10−6 4.37497× 10−6 9.81321× 10−6 0.0000173826 0.00002704761.10034× 10−6 4.39221× 10−6 9.85678× 10−6 0.0000174685 0.0000271951.10309× 10−6 4.40535× 10−6 9.89118× 10−6 0.0000175381 0.000027317

Finally, we point out that, for given equations with initial valuesu(x,0), we may increase the accuracy of tseries solution by computing more terms which is quite easy using one of the symbolic programming pMATHEMATICA , MATLAB , . . . , etc.

The solutions are very rapidly convergent by utilizing the ADM. The numerical results we obtained justadvantage of this methodology. Furthermore, as the decomposition method does not require discretizativariables, i.e., time and space, it is not effected by computation round off errors and necessity of large cmemory and time. Clearly, the series solution methodology can be applied to various type of linear or noordinary differential equations[17,18]and partial differential equations[19–29]as well.

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