Mathematical Problems in Engineering

Volume 2009, Article ID 737928, 13 pages

http://dx.doi.org/10.1155/2009/737928

## Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term

^{1}Department of Mathematics, Universidad de Caldas, Campus La Nubia, Manizales, Caldas Cll 65 # 26-10, A. A. 275, Colombia^{2}Department of Mathematics and Statistics, National University of Colombia, Manizales Cll 65, Palogrande Stadium, Colombia^{3}Department of Mathematics, National University of Colombia, Bogotá, Colombia

Received 20 April 2009; Revised 14 August 2009; Accepted 31 August 2009

Academic Editor: Ji Huan He

Copyright © 2009 Alvaro H. Salas S and Cesar A. Gómez S. 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.

#### Abstract

The general projective Riccati equation method and the Exp-function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term.

#### 1. Introduction

It is well known that in an early phase of the development of the solitons theory, there were already many applications in physics and engineering. In particular, traveling waves as solutions of the KdV equation

have been of some interest since 150 years. Some generalizations of this last equation have been studied recently. For instance, the equation

where , are arbitrary constants, which have applications in physics, has been analyzed in [1] from the point of view of its exact solutions. The search of explicit solutions to nonlinear partial differential equations (NLPDEs) using analytic methods is not an easy task. However, the use of computational methods facilitates this work. Some powerful computational methods such as the tanh method [2], the generalized tanh method [3, 4], the extended tanh method [5–10], the improved tanh-coth method [11–13], the Exp-function method [14–18], the modified Exp-function method [19], the Cole-Hopf transformation [20], the projective Riccati equation method (PREM) [21, 22], the generalized projective Riccati equations method [23–25], the extended hyperbolic function method [26], and many other methods have been developed in this direction. The PREM and the Exp-function method have been used in a satisfactory form to solve some NLPDEs [15–17, 24, 25, 27–30]. In this paper, we use this last two methods to obtain soliton and periodic solutions to the following special KdV equation with variable coefficients and forcing term:

where is an external forcing function varying with time , is a constant, and is a function of , . Equation (1.3) is a generalization of the following equation [15, 17, 31]:

which results from (1.3) by taking and .

We suppose that the solution to (1.3) has the form

Therefore, (1.3) reduces to

Now we consider the transformation

where is a constant and is an unknown function of to be determined later. Substituting (1.7) into (1.6), we obtain

where

#### 2. The Exp-Function Method

Recently He and Wu [14] have introduced the Exp-function method to solve nonlinear equations. In particular, the Exp-function method is a useful tool 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 [14–19]. The Exp-function method is very simple and straightforward and is based on a priori assumption that traveling wave solutions to a nonlinear partial differential equation in the form

can be found using the expression

where , , , and are positive integers which could be freely chosen; and are unknown constants to be determined. According to this, we suppose that solutions to (1.8) can be expressed in the form

where , , , and are positive integers which are unknown to be determined later; and are unknown constants. We have the following two cases.

##### 2.1. Case 1: and

In this case, the trial solution to (1.8) becomes

Substituting (2.4) into (1.8) we obtain a polynomial equation in the variable . Equating to zero the coefficients of all powers of yields a set of algebraic equations. Solving it with the aid of a computer, we get , , , and , and, from (1.5), (1.7), and (2.3), one solution to (1.3) is given by

where

and , , and are arbitrary real or complex numbers.

##### 2.2. Case 2: and

In this case, the trial solution to (1.8) becomes

As in the first case, we obtain an algebraic system. Solving it gives

From (1.5), (1.7), and (2.7) we may verify that to this set of values corresponds the solution

where

and , , , and are arbitrary real or complex numbers.

#### 3. General Projective Riccati Equation Method

The projective Riccati equation method was introduced initially in [21] and generalizations of this method have been used in a satisfactory way by several authors to solve nonlinear partial equations [22–25]. Using this last method [24, 25, 27–30], we seek solutions to (1.8) in the form

where , , are constants and and satisfy the system

In (3.2), and and are certain constants. These equations have following solutions.

*Case 1. * When and ,

*Case 2. * When and ,

*Case 3. * When ,
In this last case, we seek solutions to (1.8) in the form
where .

For any pair of functions given by (3.3) or (3.4) the following equation holds:

##### 3.1. Periodic and Soliton Solutions

Periodic and soliton solutions are obtained when and and it corresponds to the first two cases. Substituting (3.1), along with (3.2) and (3.7) into the left hand of (1.8) and collecting all terms with the same power in , we get a polynomial in the variables and . We may choose . Thus, we seek solutions to (1.8) in the form

We equate each coefficient of the polynomial to zero. This will give an overdetermined system of algebraic equations involving the parameters , , and , , and the unknown function . Having determined these parameters, we may determine , and using (1.5) we obtain an exact solution in a closed form. The corresponding system reads.

(i)(ii)(iii)(iv)(v), (vi)(vii)(viii)(ix)(x)In the equations above, , , and . Solving the previous system with the aid of a computer, we obtain many solutions to (1.3). These solutions may be obtained from (1.5) and are given by (3.9)–(3.19). In these formulas, and , are arbitrary parameters. In a formula containing , we suppose that and if the expression appears, we choose . If a formula involves and (see, e.g., (3.17)) we consider that and

*First Group. :. *

(i), , , :(ii), , , :(iii), , , , , :(iv), , , , , :
Figure 1 shows solution for the choices , , , , , , , , and :
Figure 2 shows solution for the choices , , , , , , , , and .

*Second Group. :. *

(v), , , , :(vi), , , , :(vii), , , , :(viii), , , , , :

##### 3.2. Rational Solutions

We seek rational solutions to (1.8) () in the form given by (3.6) with ,

where

Note that this last equation has the general solution

We now substitute (3.20) into (1.8), and using the relation we obtain an equation whose left-hand side is a polynomial in the variable . We equate each coefficient of this polynomial to zero and we get the following algebraic system.

(i)(ii)(iii)(iv)Solving this system gives , , and

A rational solution of (1.8) is given by

According to (1.5), (1.6), and (1.7) we obtain the following rational solution to (1.3):

#### 4. Conclusions

In this paper, by using the projective Riccati equation and the Exp-function methods, with the help of a symbolic computation engine, we obtain exact solutions for a generalized KdV equation with forcing term (1.3). The methods certainly works well for a large class of very interesting nonlinear equations. The main advantage of these methods is their capability of greatly reducing the size of computational work compared to existing techniques such as the pseudospectral method, the inverse scattering method, Hirota's bilinear method, and the truncated Painlevé expansion. The Exp-function method gives us more general solutions with some free parameters than the projective Riccati equation method. It also has other interesting applications. For instance, the Exp-function method may be applied not only to differential equations but also to differential-difference equations or Stochastic equations [32–35].

#### References

- N. Nirmala, M. J. Vedan, and B. V. Baby, “Auto-Bäcklund transformation, Lax pairs, and Painlevé property of a variable coefficient Korteweg-de Vries equation,”
*Journal of Mathematical Physics*, vol. 27, no. 11, pp. 2640–2646, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Fan and Y. C. Hon, “Generalized tanh method extended to special types of nonlinear equations,”
*Zeitschrift fur Naturforschung A*, vol. 57, no. 8, pp. 692–700, 2002. View at Google Scholar · View at Scopus - C. A. Gómez, “Exact solutions for a new fifth-order integrable system,”
*Revista Colombiana de Matemáticas*, vol. 40, no. 2, pp. 119–125, 2006. View at Google Scholar · View at MathSciNet - C. A. Gómez and A. H. Salas, “Exact solutions for a reaction diffusion equation by using the generalized tanh method,”
*Scientia et Technica*, vol. 13, no. 35, pp. 409–410, 2007. View at Google Scholar - C. A. Gómez, A. H. Salas, and B. Acevedo Frias, “New periodic and soliton solutions for the generalized BBM and Burger's-BBM equations,”
*Applied Mathematics and Computation*. In press. - C. A. Gómez and A. H. Salas, “Exact solutions for a new integrable system (KdV6),”
*Journal of Mathematical Sciences: Advances and Applications*, vol. 1, no. 2, pp. 401–413, 2008. View at Google Scholar · View at MathSciNet - A. H. Salas, C. A. Gómez, and G. Escobar, “Exact solutions for the general fifth order KdV equation by the extended tanh method,”
*Journal of Mathematical Sciences: Advances and Applications*, vol. 1, no. 2, pp. 305–310, 2008. View at Google Scholar · View at MathSciNet - C. A. Gómez, “A new travelling wave solution of the Mikhailov-Novikov-Wang system using the extended tanh method,”
*Boletín de Matemáticas*, vol. 14, no. 1, pp. 38–43, 2007. View at Google Scholar · View at MathSciNet - A.-M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations,”
*Applied Mathematics and Computation*, vol. 184, no. 2, pp. 1002–1014, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Gómez, “Special forms of the fifth-order KdV equation with new periodic and soliton solutions,”
*Applied Mathematics and Computation*, vol. 189, no. 2, pp. 1066–1077, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Gómez and A. H. Salas, “The generalized tanh-coth method to special types of the fifth-order KdV equation,”
*Applied Mathematics and Computation*, vol. 203, no. 2, pp. 873–880, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Salas and C. A. Gómez, “Computing exact solutions for some fifth KdV equations with forcing term,”
*Applied Mathematics and Computation*, vol. 204, no. 1, pp. 257–260, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Gómez and A. H. Salas, “The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation,”
*Applied Mathematics and Computation*. In press. View at Publisher · View at Google Scholar - J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,”
*Chaos, Solitons & Fractals*, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Zhang, “Exp-function method exactly solving the KdV equation with forcing term,”
*Applied Mathematics and Computation*, vol. 197, no. 1, pp. 128–134, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He and L.-N. Zhang, “Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method,”
*Physics Letters A*, vol. 372, no. 7, pp. 1044–1047, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - S. Zhang, “Application of Exp-function method to a KdV equation with variable coefficients,”
*Physics Letters A*, vol. 365, no. 5-6, pp. 448–453, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - A. H. Salas, “Exact solutions for the general fifth KdV equation by the exp function method,”
*Applied Mathematics and Computation*, vol. 205, no. 1, pp. 291–297, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Salas, C. A. Gómez, and J. Castillo, “New abundant solutions for the Burger's equation,”
*Computers & Mathematics with Applications*, vol. 58, pp. 514–520, 2009. View at Google Scholar - C. A. Gómez and A. H. Salas, “The Cole-Hopf transformation and improved tanh-coth method applied to new integrable system (KdV6),”
*Applied Mathematics and Computation*, vol. 204, no. 2, pp. 957–962, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Conte and M. Musette, “Link between solitary waves and projective Riccati equations,”
*Journal of Physics. A*, vol. 25, no. 21, pp. 5609–5623, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Salas, “Some solutions for a type of generalized Sawada-Kotera equation,”
*Applied Mathematics and Computation*, vol. 196, no. 2, pp. 812–817, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Yan, “The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations,”
*MMRC, AMSS, Academis Sinica*, vol. 22, pp. 275–284, 2003. View at Google Scholar - E. Yomba, “The general projective Riccati equations method and exact solutions for a class of nonlinear partial differential equations,”
*Chinese Journal of Physics*, vol. 43, no. 6, pp. 991–1003, 2005. View at Google Scholar · View at MathSciNet - C. A. Gómez and A. H. Salas, “Special forms of Sawada-Kotera equation with periodic and soliton solutions,”
*International Journal of Applied Mathematical Analysis and Applications*, vol. 3, no. 1, pp. 45–51, 2008. View at Google Scholar · View at MathSciNet - Y. Shang, Y. Huang, and W. Yuan, “New exact traveling wave solutions for the Klein-Gordon-Zakharov equations,”
*Computers & Mathematics with Applications*, vol. 56, no. 5, pp. 1441–1450, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Gómez and A. H. Salas, “Exact solutions for the generalized shallow water wave equation by the general projective Riccati equations method,”
*Boletín de Matemáticas*, vol. 13, no. 1, pp. 50–56, 2006. View at Google Scholar · View at MathSciNet - C. A. Gómez and A. H. Salas, “New exact solutions for the combined sinh-cosh-Gordon equation,”
*Lecturas Matematicas*, vol. 27, pp. 87–93, 2006. View at Google Scholar - C. A. Gómez, “New exact solutions of the Mikhailov–Novikov–Wang system,”
*International Journal of Computer, Mathematical Sciences and Applications*, vol. 1, pp. 137–143, 2007. View at Google Scholar - Y. Chen and B. Li, “General projective Riccati equation method and exact solutions for generalized KdV-type and KdV-Burgers-type equations with nonlinear terms of any order,”
*Chaos, Solitons & Fractals*, vol. 19, no. 4, pp. 977–984, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Taogetusang and Sirendaoerji, “The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term,”
*Chinese Physics*, vol. 15, no. 12, pp. 2809–2818, 2006. View at Publisher · View at Google Scholar · View at Scopus - S.-D. Zhu, “Exp-function method for the Hybrid-Lattice system,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 3, pp. 461–464, 2007. View at Google Scholar · View at Scopus - S.-D. Zhu, “Exp-function method for the discrete mKdV lattice,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 3, pp. 465–468, 2007. View at Google Scholar · View at Scopus - C. Dai, X. Cen, and S. S. Wu, “The application of He's exp-function method to a nonlinear differential-difference equation,”
*Chaos, Solitons & Fractals*, vol. 41, no. 1, pp. 511–515, 2009. View at Publisher · View at Google Scholar · View at Scopus - C.-Q. Dai and J.-F. Zhang, “Application of he's EXP-function method to the stochastic mKdV equation,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 10, no. 5, pp. 675–680, 2009. View at Google Scholar · View at Scopus