Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 498249 | 10 pages | https://doi.org/10.1155/2010/498249

Exact Solutions for the Generalized BBM Equation with Variable Coefficients

Academic Editor: Jihuan He
Received23 Nov 2009
Accepted21 Jan 2010
Published15 Mar 2010

Abstract

The variational iteration algorithm combined with the exp-function method is suggested to solve the generalized Benjamin-Bona-Mahony equation (BBM) with variable coefficients. Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered.

1. Introduction

The BBM equation

𝑢𝑡+𝑢𝑢𝑥+𝑢𝑥𝜇𝑢𝑥𝑥𝑡=0,(1.1) which describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems, has been proposed by Benjamin et al. in 1972 [1] as a more satisfactory model than the KdV equation [2]

𝑢𝑡+𝑢𝑢𝑥+𝑢𝑥𝑥𝑥=0.(1.2) It is easy to see that (1.1) can be derived from the equal width EW-equation [3]:

𝑢𝑡+𝑢𝑢𝑥𝜇𝑢𝑥𝑥𝑡=0,(1.3) by means of the change of variable 𝑢=𝑢+1, that is, by replacing 𝑢 with 𝑢+1. This last equation is considered as an equally valid and accurate model for the same wave phenomena simulated by (1.1) and (1.2). On the other hand, some researches analyzed the generalized KdV equation with variable coefficients

𝑢𝑡+𝜎(𝑡)𝑢𝑝𝑢𝑥+𝜇(𝑡)𝑢𝑥𝑥𝑥=0,(1.4) because this model has important applications in several fields of science [47].

Motivated by these facts, we will consider here the generalized EW-equation with variable coefficients

𝑢𝑡+𝜎(𝑡)𝑢𝑝𝑢𝑥𝜇(𝑡)𝑢𝑥𝑥𝑡=0.(1.5) Using the solutions of (1.5) we obtain exact solutions to the generalized BBM equation

𝑢𝑡+𝜎(𝑡)(𝑢+1)𝑝𝑢𝑥𝜇(𝑡)𝑢𝑥𝑥𝑡=0,(1.6) of order 𝑝>0.

2. Exact Solutions to Generalized BBM Equation

2.1. The Variational Iteration Method

Consider the following nonlinear equation:

𝐿𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)=𝑔(𝑥,𝑡),(2.1) where 𝐿 and 𝑁 are linear and nonlinear operators, respectively, and 𝑔(𝑥,𝑡) is an inhomogeneous term. According to the variational iteration method (VIM) [814], a functional correction to (2.1) is given by

𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)+𝑡0𝜃(𝜏)𝐿𝑢𝑛(𝑥,𝜏)+𝑁̃𝑢𝑛(𝑥,𝜏)𝑔(𝑥,𝜏)𝑑𝜏,(2.2) where 𝜃(𝜏) is a general Lagrange's multiplier, which can be identified via the variational theory; the subscript 𝑛0 denotes the 𝑛th order approximation and ̃𝑢 is a restricted variation which means 𝛿̃𝑢=0. In this method, we first determine the Lagrange multiplier 𝜃(𝜏) that will be identified optimally via integration by parts. The successive approximation 𝑢𝑛+1 of the solution 𝑢 will be readily obtained upon using the determined Lagrangian multiplier and any selective function 𝑢0. One of the advantages of the VIM, is the free choice of the initial solution 𝑢0(𝑥,𝑡). If we consider a special form to 𝑢0 with arbitrary parameters, using the relations

𝑢𝑛(𝑥,𝑡)=𝑢𝑛+1𝜕(𝑥,𝑡),𝑘𝜕𝑡𝑘𝑢𝑛𝜕(𝑥,𝑡)=𝑘𝜕𝑡𝑘𝑢𝑛+1(𝑥,𝑡),(2.3) we can obtain a set of algebraic equations in the unknowns given by the parameters that appear in 𝑢0. Solving this system, we have exact solutions to (2.1). To solve (1.5), we construct the following functional equation

𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)+𝑡0𝜃(𝜏)𝐿𝑢𝑛(𝑥,𝜏)+𝑁̃𝑢𝑛(𝑥,𝜏)𝑑𝜏,(2.4) where

𝐿𝑢𝑛𝑢(𝑥,𝜏)=𝑛𝜏(𝑥,𝜏),𝑁̃𝑢𝑛(𝑥,𝜏)=𝜎(𝜏)(̃𝑢+1)𝑝̃𝑢𝑥(𝑥,𝜏)𝜇(𝜏)̃𝑢𝑥𝑥𝜏(𝑥,𝜏).(2.5) Taking in (2.4) variation with respect to the independent variable 𝑢𝑛, and noticing that 𝛿𝑁̃𝑢𝑛=0 we have

𝛿𝑢𝑛+1(𝑥,𝑡)=𝛿𝑢𝑛(𝑥,𝑡)+𝛿𝑡0𝜃(𝜏)𝐿𝑢𝑛(𝑥,𝜏)+𝑁̃𝑢𝑛(𝑥,𝜏)𝑑𝜏=𝛿𝑢𝑛(𝑥,𝑡)+𝜃(𝑡)𝛿𝑢𝑛(𝑥,𝑡)𝑡0𝜃(𝜏)𝛿𝑢𝑛(𝑥,𝜏)𝑑𝜏=0.(2.6) This yields the stationary conditions

𝜃1+𝜃(𝑡)=0,(𝑡)=0.(2.7) Therefore,

𝜃(𝑡)=1.(2.8) Substituting this value into (2.4) we obtain the formula

𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)𝑡0𝐿𝑢𝑛(𝑥,𝜏)+𝑁̃𝑢𝑛(𝑥,𝜏)𝑑𝜏.(2.9)

Using the wave transformation

𝜉=𝑥+𝜆𝑡+𝜉0,(2.10) setting

𝜕𝑢𝜕𝑡1𝜕(𝜉)=𝑢𝜕𝑡0(𝜉),(2.11) and performing one integration, (2.9) reduces to

𝜆𝑢0(𝜉)+𝜎(𝑡)𝑢𝑝+10𝑝+1(𝜉)𝜆𝜇(𝑡)𝑢0(𝜉)=0,(2.12) where for sake of simplicity we set the constant of integration equal to zero. With the change of variable

𝑢0(𝜉)=𝑣2/𝑝(𝜉),(2.13) equation (2.12) converts to

𝜆𝑣2(𝜉)2𝜇(𝑡)(2𝑝)𝑝2𝜆𝑣22𝜇(𝑡)𝑝𝜆𝑣(𝜉)𝑣(𝜉)+𝜎(𝑡)𝑝+1𝑣(𝜉)4=0.(2.14) Observe that if 𝑣(𝜉) is a solution to (2.14), then 𝑣(𝜉) is also a solution to this equation.

2.2. The Exp-Function Method

Recently, He and Wu [15] have introduced the Exp-function method to solve nonlinear differential equations. In particular, the Exp-function method is an effective method for solving nonlinear equations with high nonlinearity. The method has been used in a satisfactory way by other authors to solve a great variety of nonlinear wave equations [1521]. The Exp-function method is very simple and straightforward, and can be briefly revised as follows: Given the nonlinear partial differential equation

𝐹𝑢,𝑢𝑥,𝑢𝑡,𝑢𝑥𝑥,𝑢𝑥𝑡,𝑢𝑡𝑡,=0,(2.15) it is transformed to ordinary differential equation

𝐹𝑢,𝑢,𝑢,𝑢,𝑢𝑥𝑡,=0,(2.16) by mean of wave transformation 𝜉=𝑥+𝜆𝑡+𝜉0. Solutions to (2.16) can then be found using the expression

𝑢(𝜉)=𝑑𝑛=𝑐𝑎𝑛exp(𝑛𝜉)𝑞𝑛=𝑝𝑏𝑛,exp(𝑛𝜉)(2.17) where 𝑐,𝑑,𝑝, and 𝑞 are positive integers which are unknown to be determined later, 𝑎𝑛 and 𝑏𝑛 are unknown constants.

After balancing, we substitute (2.17) into (2.16) to obtain an algebraic systems in the variable 𝜁=exp(𝑛𝜉). Solving the algebraic system we can obtain exact solutions to (2.16) and reversing, solutions to (2.15) in the original variables.

3. Solutions to (2.14) by the Exp-Function Method

Using the Exp-function method, we suppose that solutions to (2.14) can be expressed in the form

𝑣(𝜉)=1𝑛=1𝑎𝑛exp(𝑛𝑟𝜉)1𝑚=1𝑏𝑚=𝑎exp(𝑚𝑟𝜉)1exp(𝑟𝜉)+𝑎0+𝑎1exp(𝑟𝜉)𝑏1exp(𝑟𝜉)+𝑏0+𝑏1exp(𝑟𝜉).(3.1) We obtain following solutions to (2.14): 𝑣1=±2𝜆𝑘(𝑝+1)(𝑝+2)2(𝑝+1)(𝑝+2)𝜆exp𝑝/2𝜉𝜇(𝑡)𝑘2𝜎(𝑡)exp𝑝/2𝜉𝑣𝜇(𝑡),𝜆=𝜆(𝑡),2=±2𝜆𝑘(𝑝+1)(𝑝+2)𝜎(𝑡)𝑘2exp𝑝/2𝜉𝜇(𝑡)2𝜆(𝑝+1)(𝑝+2)exp𝑝/2𝜉𝜇(𝑡),𝜆=𝜆(𝑡).(3.2) Some special solutions are obtained if

𝑘𝜆=𝜆(𝑡)=±22𝑝2+3𝑝+2𝜎(𝑡).(3.3) This choice gives solutions

𝑣3𝑘=±2𝑝csch2𝜉𝑘𝜇(𝑡),𝜆=22𝑝2𝑣+3𝑝+2𝜎(𝑡),(3.4)4=𝑘2𝑝sech2𝜉𝑘𝜇(𝑡),𝜆=22𝑝2𝑣+3𝑝+2𝜎(𝑡),(3.5)5𝑘=±2𝑝csc2𝜇𝜉𝑘(𝑡),𝜆=22𝑝2𝜎+3𝑝+2(𝑡).(3.6) Solution (3.6) follows from (3.4) with the identifications 𝜇(𝑡)𝜇(𝑡) and 𝑘𝑘1.

𝑣6𝑘=2𝑝sec2𝜉𝑘𝜇(𝑡),𝜆=22𝑝2+3𝑝+2𝜎(𝑡).(3.7) Solution (3.7) follows from (3.4) with the identifications 𝜇(𝑡)𝜇(𝑡) and 𝑘𝑘.

4. Particular Cases

4.1. Case 1: Solutions to (2.14) When 𝑝=2

Equation (2.14) takes the form

𝜆𝑣2(𝜉)𝜆𝜇(𝑡)𝑣(𝜉)𝑣1(𝜉)+3𝜎(𝑡)𝑣(𝜉)4=0.(4.1) From (3.2) with 𝑝=2:

𝑣7=±24𝜆𝑘24𝜆exp1/𝜇𝜉(𝑡)𝑘2𝜎(𝑡)exp1/𝜇𝜉,𝑣(𝑡)8=±24𝜆𝑘𝑘2𝜎(𝑡)exp1/𝜉𝜇(𝑡)24𝜆exp1/𝜉.𝜇(𝑡)(4.2) From (3.3)–(3.7) with 𝑝=2:

𝑣9𝑘=±21csch𝜉𝑘𝜇(𝑡),𝜆=2𝑣24𝜎(𝑡),10𝑘=±21sech𝜉𝑘𝜇(𝑡),𝜆=2𝑣24𝜎(𝑡),11𝑘=±21csc𝜉𝑘𝜇(𝑡),𝜆=2𝜎𝑣24(𝑡),12𝑘=±21sec𝜉𝑘𝜇(𝑡),𝜆=224𝜎(𝑡).(4.3) Other exact solutions are:

𝑣13=±3𝑎22exp2/𝜇(𝑡)𝜉+255𝑎exp𝑘2/𝜇(𝑡)𝜉223𝑎22exp2/𝜇(𝑡)𝜉+22𝑎exp12/𝜇(𝑡)𝜉+22,𝜆=3𝑘2𝑣𝜎(𝑡),14=±3𝑎2±255𝑎exp22/𝜇(𝑡)𝜉22exp𝑘2/𝜇(𝑡)𝜉3𝑎2+22𝑎exp22/𝜇(𝑡)𝜉+22exp12/𝜇(𝑡)𝜉,𝜆=3𝑘2𝑣𝜎(𝑡),15=±𝑘1448+553𝑎11+55exp2/𝜇(𝑡)𝜉+228+155,𝜆=3𝑘2𝑣𝜎(𝑡),16𝑘=±𝑎±sinh2/𝜇(𝑡)𝜉𝑎2+1±cosh12/𝜇(𝑡)𝜉,𝜆=3𝑘2𝑣𝜎(𝑡),17𝑘=±𝑎±cosh2/𝜇(𝑡)𝜉𝑎2+1±sinh12/𝜇(𝑡)𝜉,𝜆=3𝑘2𝑣𝜎(𝑡),18=±𝑘cos2/𝜇(𝑡)𝜉1±sin𝑘2/𝜇(𝑡)𝜉,𝜆=23𝜎(𝑡).(4.4)

4.2. Case 2: Solutions to (2.14) When 𝑝=4

Equation (2.14) takes the form

𝜆𝑣2𝑣(𝜉)+𝜇(𝑡)𝜆2𝜆2𝜇(𝑡)𝑣(𝜉)𝑣1(𝜉)+5𝜎(𝑡)𝑣(𝜉)4=0.(4.5) From (3.2) with 𝑝=4:

𝑣19=±60𝜆𝑘60𝜆exp2/𝜇𝜉(𝑡)𝑘2𝜎(𝑡)exp2/𝜇𝜉𝑣(𝑡),𝜆=𝜆(𝑡),20=±60𝜆𝑘𝜎(𝑡)𝑘2exp2/𝜉𝜇(𝑡)60𝜆exp2/𝜉𝜇(𝑡),𝜆=𝜆(𝑡).(4.6) From (3.3)–(3.7) with 𝑝=4:

𝑣21𝑘=±22csch𝜉𝑘𝜇(𝑡),𝜆=2𝑣60𝜎(𝑡),22𝑘=±22sech𝜉𝑘𝜇(𝑡),𝜆=2𝑣60𝜎(𝑡),23𝑘=±22csc𝜉𝑘𝜇(𝑡),𝜆=2𝜎𝑣60(𝑡),24𝑘=±22sec𝜉𝑘𝜇(𝑡),𝜆=260𝜎(𝑡).(4.7) Other exact solutions are:

𝑣25𝑘=±𝑎exp2/𝜉𝜇(𝑡)42𝑎2exp4/𝜉𝜇(𝑡)+16𝑎exp2/𝜉1𝜇(𝑡)+16,𝜆=5𝑘2𝑣𝜎(𝑡),26𝑘=±4exp2/𝜇(𝑡)𝜉𝑎2𝑎2+16𝑎exp2/𝜉𝜇(𝑡)+16exp4/𝜉1𝜇(𝑡),𝜆=5𝑘2𝑣𝜎(𝑡),273=±2𝑘12±cos2/𝜉4𝜇(𝑡),𝜆=5𝑘2𝑣𝜎(𝑡),283=±2𝑘12±sin2/𝜉4𝜇(𝑡),𝜆=5𝑘2𝜎(𝑡).(4.8) It is clear that using (2.13) we obtain solutions to (1.5). Finally, observe that if 𝑢0(𝑥,𝑡) is a solution of (1.5), then the solutions 𝑢(𝑥,𝑡) to the generalized BBM equation (1.6) are obtained as follows:

𝑢(𝑥,𝑡)=𝑢0(𝑥,𝑡)1.(4.9)

5. Conclusions

We have considered the generalized EW-equation with variable coefficients and the generalized BBM-equation with variable coefficients. We obtained analytic solutions by using the variational iteration method combined with the exp-function method. With the aid of Mathematica we have derived a lot of different types of solutions for these two models. Combined formal soliton-like solutions as well as kink solutions have been formally derived. The results obtained show that the technique used here can be considered as a powerful method to analyze other types of nonlinear wave equations.

According to [22], there are alternative iteration alorithms, which might be useful for future work. Furthermore, various modifications of the exp-function method have been appeared in open literature, for example, the double exp-function method [23, 24].

Other methods for solving nonlinear differential equations may be found in [2535].

We think that the results presented in this paper are new in the literature.

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Copyright © 2010 Cesar A. Gómez and Alvaro H. Salas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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