On experimental results and explicit exact solutions for the generalized Boussinesq type equation

Applied Mathematics and Computation 165 (2005) 303–311

www.elsevier.com/locate/amc

On experimental results and explicitexact solutions for the generalized

Boussinesq type equation

Dogan Kaya

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

Abstract

We implemented a decomposition method for approximating the solution of the gen-

eralized Boussinesq type equation. By using this scheme, the explicit exact solution is

calculated in the form of a convergent power series with easily computable components.

To illustrate the application 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 Boussinesq type equation; Traveling wave

solution; Solitary wave solution

1. Introduction

In this study, we consider the initial value problem for the generalized Bous-

sinesq type equation

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

doi:10.1016/j.amc.2004.04.049

E-mail address: [email protected]

mailto:[email protected]

304 D. Kaya / Appl. Math. Comput. 165 (2005) 303–311

utt � uxx þ uxxxx ¼ �ðwðuÞÞxx; x 2 R; t > 0;

uðx; 0Þ ¼ f ðxÞ; and utðx; 0Þ ¼ gðxÞ;ð1Þ

where w(u) = juja � 1u and a > 1. This equation represent a generalization of the

classical Boussinesq equation which arises in the modeling of nonlinear strings.

Eq. (1) describes in the continuous limit the propagation of waves in a one-

dimensional nonlinear lattice and the propagation of waves in shallow water

[1–3]. A proof of the local well-posedness for (1) has been shown by Bona

and Sachs [4]. In [5] the authors noticed that Eq. (1) admits solitary solutions

and existence of solitary wave solutions illustrates the perfect balance betweenthe dispersion and the nonlinearity of Eq. (1). A specific solution of Eq. (1) are

given explicitly as

uðx; tÞ ¼ Ksech2

a�1ðBnÞ; ð2Þ

where a is an integer, K ¼ ðaþ1Þð1�c2Þ2

h i 1a�1

, B ¼ ð1�c2Þ1=2ða�1Þ2

, n = x � ct, and c is the

wave speed satisfying c2 < 1 (see [3]). For a integer, the solitary-wave solutions

have been shown to be stable under some restrictions on the wave speed by

Bona and Sachs [4].

The aim of this paper is the study of the exact and numerical solutions of the

Eq. (1). Finding explicit exact and numerical solutions of nonlinear equations

efficiently is of major importance and has widespread applications in numericalmethods 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 generalized Boussinesq type equation for a given

nonlinearity (w(u))xx.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. As numerical tool the method provide us with numerical

solution without discretization of the given equation. The writing of the com-

puter codes are straightforward in any symbolic programming language. 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 sim-

plicity into superficial conclusions [7].

In this paper, various Boussinesq equations [4,5] can be handled more

easily, quickly, and elegantly by implementing the ADM rather thanthe traditional methods for finding the analytical as well as numerical solu-

tions.

D. Kaya / Appl. Math. Comput. 165 (2005) 303–311 305

2. Analysis of the method

In this section we outlined the steps to obtain analytic solution of Boussin-

esq equation (1) using the ADM. First we write the Boussinesq equation in a

standard operator form

Ltu� uxx þ Lxuþ ðwðuÞÞxx ¼ 0; ð3Þ

where the notations Lt ¼ o2

ot2 and Lx ¼ o4

ox4 symbolize the linear differential oper-

ators. Assuming the inverse of the operator L�1t exists and it can conveniently

be taken as the twofold integration inverse operator L�1t . Thus, applying the

inverse operator L�1t to (2) yields

L�1t Ltu ¼ L�1

t ðuxx � Lxu� ðwðuÞÞxxÞ: ð4Þ

Therefore, it follows that

uðx; tÞ ¼ uðx; 0Þ þ tutðx; 0Þ þ L�1t ðuxx � Lxu� ðwðuÞÞxxÞ: ð5Þ

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

by the series

uðx; tÞ ¼X1n¼0

unðx; tÞ: ð6Þ

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

conditions, i.e.,

u0 ¼ uðx; 0Þ þ tutðx; 0Þ: ð7ÞThe remaining components un(x, t), nP 1, can be completely determined

such that each term is computed by using the previous term. Since u0 is known,

un ¼ L�1t ððun�1Þxx � Lxun�1 � ðAn�1ÞxxÞ; nP 1; ð8Þ

where wðuÞ ¼P1

n¼0Anðu0; u1; . . . ; unÞ. The components An are called the

Adomian polynomials, these polynomials can be calculated for all forms of

nonlinearity according to specific algorithms constructed by Adomain [6,9].In this specific nonlinearity, we use the general form of formula for An polyno-

mials as

An ¼1

n!dn

dknwX1k¼0

kkuk

!" #k¼0

; nP 0: ð9Þ

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.

The first few Adomian polynomials for the nonlinearity f(u) are


A0 ¼ wð1Þðu0Þ; A1 ¼ u1wð1Þðu0Þ; A2 ¼ u2w

ð1Þðu0Þ þ1

2!u21w

ð2Þðu0Þ;

A3 ¼ u3wð1Þðu0Þ þ u2u1w

ð2Þðu0Þ þ1

3!u31w

ð3Þðu0Þ

and so on, the rest of the polynomials can be constructed in a similar manner.

To follow of the reader, some of the first Adomian polynomials An are com-

puted according to (9) with w(u) = ua and a >1 and this gives

A0 ¼ ua0; A1 ¼ aua�10 u1; A2 ¼ ða� 1Þaua�2

0

1

2!u21 þ aua�1

0 u2;

A3 ¼ ða� 2Þða� 1Þaua�30

1

3!u31 þ ða� 1Þaua�2

0 u1u2 þ aua�10 u3;

and so on, the rest of the polynomials can be constructed in a similar manner.

A slight modification to the ADM was proposed by Wazwaz [8] that gives

some flexibility in the choice of the zeroth component u0 to be any simple

term and modify the term u1 accordingly and 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

u0 ¼ uðx; 0Þ; u1 ¼ tutðx; 0Þ þ L�1t ððu0Þxx � Lxu0 � ðA0ÞxxÞ;

un ¼ L�1t ððun�1Þxx � Lxun�1 � ðAn�1ÞxxÞ; nP 2: ð10Þ

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

limn!1

/n ¼ uðx; tÞ; where /nðx; tÞ ¼Xnk¼0

ukðx; tÞ; nP 0 ð11Þ

and the recurrence relation is given as in (10).

Numerical computations of the generalized Boussinesq equation (1) haveoften been repeated in the literature. However, to show the effectiveness of

the proposed decomposition method and to give a clear overview of the meth-

odology some examples of the generalized Boussinesq equation (1) will be dis-

cussed in the following section.

3. Numerical implementation for generalized Boussinesq equation

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

eralized Boussinesq type equation (1). Existence and derivations of such solu-

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

In the first example, we will consider Eq. (1) for the special case associated

with the initial conditions


uðx; 0Þ ¼ Ksech2

a�1ðBxÞ; utðx; 0Þ ¼2BcKsechðBxÞ1þ

2a�1 sinhðBxÞ

a� 1; ð12Þ

where a is integer, K ¼ ½ðaþ1Þð1�c2Þ2

�1

a�1, B ¼ ð1�c2Þ1=2ða�1Þ2

.To find the solution of the initial value problem (1) with (12) for a = 3 we

apply the scheme (10). Performing the integration we obtain the following

u0 ¼ KsechðBxÞ; ð13Þ

u1 ¼B2Kt2

16ð�5� 115B2 þ 60K2 � ð4� 76B2 þ 36K2Þ coshð2BxÞ

þ ð1� B2Þ coshð4BxÞÞsech5ðBxÞ þ BcKtsechðBxÞ tanhðBxÞ; ð14Þ

u2 ¼B4Kt4

3072ð387þ 26010B2 þ 2337507B4 � 10368K2 � 1423008B2K2

þ 125712K4 þ ð392þ 6368B2 � 2485288B4 � 3024K2

þ 1485360B2K2 � 121536K4Þ coshð2BxÞ � ð68 coshð4BxÞ þ 18200B2

� 331612B4 � 6912K2 þ 185184B2K2 � 10800K4Þ coshð4BxÞ� ð72� 1440B2 þ 6552B4 þ 432K2 � 3024B2K2Þ coshð6BxÞþ ð1� 2B2 þ B4Þ coshð8BxÞÞsech9ðBxÞ

þ B3cKt3

48ð�21� 723B2 þ 372K2 � ð20� 236B2 þ 108K2Þ coshð2BxÞ

þ ð1� B2Þ coshð4BxÞÞsech5ðBxÞ tanhðBxÞ; ð15Þ

u3 ¼B4c2K3t4

16ð11� 9 coshð2BxÞÞð�5þ coshð2BxÞÞsechðBxÞ7

þ B6Kt6

1474560ð�81666� 13246506B2 � 1860507558B4

� 371644257582B6 þ 4473768K2 þ 956049744B2K2

þ 238080877320B4K2 � 78345072K4 � 27978831504B2K4

þ 957999744K6 � ð90648þ 7868808B2 � 325359576B4

� 477020564424B6 � 2813304K2 þ 146259888B2K2

þ 303708329160B4K2 � 6073344K4 � 34868602368B2K4

þ 1141639488K6Þ coshð2BxÞ þ ð12975 þ 9908019B2

þ 1678312845B4 � 121383780207B6 � 3160416K2

� 857138880B2K2 þ 75754321440B4K2 þ 68535360K4

� 8041942080B2K4 þ 225141120K6Þ coshð4BxÞ


þ ð28900þ 3594444B2 � 480382980B4 þ 10708911188B6

� 1226580K2 þ 233109192B2K2 � 6424260468B4K2

� 15353856K4 þ 575350272B2K4 � 10922688K6Þ coshð6BxÞ

þ ð6226� 916230B2 þ 26994198B4 � 237231970B6 þ 268728K2

� 11997840B2K2 þ 131818200B4K2 þ 529200K4

� 7786800B2K4Þ coshð8BxÞ � ð716� 19644B2 þ 77108B4

� 531428B6 þ 4644K2 � 63720B2K2 þ 245700B4K2Þ coshð10BxÞ

þ ð1� 3B2 þ 3B4 � B6Þ coshð12BxÞÞsechðBxÞ13

þ B5cKt5

15360ð3395þ 285530B2 þ 24720995B4 � 140832K2

� 16456320B2K2 þ 2046096K4 þ ð4136þ 172928B2 � 19971304B4

� 88512K2 þ 12998112B2K2 � 1506240K4Þ coshð2BxÞþ ð508� 108248B2 þ 1736668B4 þ 50592K2 � 1059456B2K2

þ 97200K4Þ coshð4BxÞ � ð232� 4352B2 þ 19672B4

þ 1728K2 � 9504B2K2Þ coshð6BxÞ

þ ð1� 2B2 þ B4Þ coshð8BxÞÞsech9ðBxÞ tanhðBxÞ; ð16Þ

in this manner the components of the decomposition series (6) are obtained as

far as we like. This series is exact to the last term, as one can verify, of the Tay-

lor series of the exact closed form solution (2) for the special case of the a = 3.

u(x, t) = K sech(B(x � ct)) [5].

In the some other examples, we will consider the generalized Boussinesq

type equation (1) with the general form of the initial conditions (12) for differ-

ent values of the a = 2,3,4,5. Again, to find the solution of this equation, wesubstitute in the scheme (10)

u0 ¼ uðx; 0Þ; ð17Þ

u1 ¼ tutðx; 0Þ þZ t

0

Z t

0

½ðu0Þxx � Lxu0 � ðA0Þxx�dtdt; ð18Þ

..

.

un ¼Z t

0

Z t

0

½ðun�1Þxx � Lxun�1 � ðAn�1Þxx�dtdt; nP 2; ð19Þ

where the Adomian polynomials An � 1 are given same as in the first example.

Performing the calculations in (10) with (9) using Mathematica and substitut-

ing into (6) gives the exact solution in a series form [2].


4. Experimental results for the generalized Boussinesq equation

The convergence of the decomposition series have investigated by several

authors. The theoretical treatment of convergence of the decomposition meth-

od has been considered in the literature [10–14]. They obtained some results

about the speed of convergence of this method providing us to solve linearand nonlinear functional equations. In recent work of Abbaoui et al. [15] have

proposed a new approach of convergence of the decomposition series. The

authors have given a new condition for obtaining convergence of the decompo-

sition series to the classical presentation of the ADM in [15].

In order to verify numerically whether the proposed methodology lead to

accurate solutions, we will evaluate the ADM solutions using the 5-term

approximation for some examples of the generalized Boussinesq type equations

Table 1

The numerical results for /n(x, t) in comparison with the analytical solution (2) when c = 0.9 for the

solitary-wave solution of Eq. (1) with various a

xi

ti 0.1 0.2 0.3 0.4 0.5

a = 2

0.1 3.85882E�14 9.11116E�12 2.27346E�10 2.24000E�09 1.32418E�08

0.2 4.16334E�14 9.21418E�12 2.24089E�10 2.17803E�09 1.27659E�08

0.3 4.26326E�14 8.90293E�12 2.10763E�10 2.01812E�09 1.17165E�08

0.4 4.97460E�14 8.20521E�12 1.88306E�10 1.77089E�09 1.01606E�08

0.5 1.01984E�14 7.16915E�12 1.58153E�10 1.45164E�09 8.19283E�09

a = 3

0.1 9.22212E�11 2.41572E�08 6.23806E�07 6.25441E�06 3.73628E�05

0.2 6.24512E�11 1.70127E�08 4.44770E�07 4.48637E�06 2.68970E�05

0.3 2.27912E�11 7.20533E�09 1.96367E�07 2.01985E�06 1.22478E�05

0.4 1.86494E�11 3.22614E�09 6.94672E�08 6.27959E�07 3.50746E�06

0.5 5.39238E�11 1.22574E�08 3.00938E�07 2.94011E�06 1.72892E�05

a = 4

0.1 1.08490E�08 2.84539E�06 7.35098E�05 7.37203E�04 4.40458E�03

0.2 3.71572E�09 1.06501E�06 2.82835E�05 2.87484E�04 1.73136E�03

0.3 4.13716E�09 9.33435E�07 2.28170E�05 2.22331E�04 1.30512E�03

0.4 9.51273E�09 2.33109E�06 5.88069E�05 5.82643E�04 3.45564E�03

0.5 1.07689E�08 2.69664E�06 6.85432E�05 6.81716E�04 4.05264E�03

a = 5

0.1 3.10382E�07 8.11248E�05 2.09347E�03 2.09829E�02 1.25325E�01

0.2 3.68011E�08 6.96360E�06 1.57153E�04 1.46270E�03 8.33436E�03

0.3 3.07825E�07 7.67091E�05 1.94752E�03 1.93600E�02 1.15060E�01

0.4 3.46934E�07 7.78636E�05 2.24304E�03 2.23599E�02 1.33111E�01

0.5 1.94362E�07 5.00396E�05 1.28427E�03 1.28361E�02 7.65352E�02


solved in the previous section. The differences between the 5-terms solution and

the exact solution for some values of the constant a are shown in Table 1. Table

1 shows that we achieved a very good approximation to the actual solution of

the equations by using only 5-terms of the decomposition series solution de-

rived above. It is evident that the overall errors can be made smaller by adding

new terms of the decomposition series.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 system of

partial differential equations [16–18] and single partial differential equations

[19–23] as well.

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Documents

On experimental results and explicit exact solutions for the generalized Boussinesq type equation