Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 218216, 11 pagesdoi:10.1155/2011/218216

Research Article

The -Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation

Hasibun Naher, Farah Aini Abdullah, and M. Ali Akbar

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Received 24 June 2011; Accepted 22 September 2011

Academic Editor: Kue-Hong Chen

Copyright © 2011 Hasibun Naher et al. 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.

We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation by the -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.

1. Introduction

The investigation of exact traveling wave solutions of nonlinear partial differential equations (NLPDEs) plays an important role in the analysis of complex physical phenomena. The NLPDEs appear in physical sciences, various scientific and engineering problems, such as, fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, chemistry and many others. In recent years, to obtain exact traveling wave solutions of NLPDEs, many effective and powerful methods have been presented in the literature, such as the Backlund transformation [1], the tanh function method [2], the extended tanh function method [3], the variational iteration method [4], the Adomian decomposition method [5, 6], the homotopy perturbation method [7], the F-expansion method [8], the Hirota’s bilinear method [9], the exp-function method [10], the Cole-Hopf transformation [11], the general projective Riccati equation method [12] and others [13–20], Nowaday, searching analytical solutions of the NLPDEs has become more lucrative partly due to the accessibility computer symbolic systems, like Maple, Mathematica, and Matlab, which help us to calculate the complicated and wearisome algebraic calculations on computer.

Recently, Wang et al. [21] introduced a new direct method called the -expansion method to seek traveling wave solutions of the nonlinear evolution equations. Abazari [22] implemented the -expansion method to NLEEs related to fluid mechanics. Zheng [23] used the method for getting exact traveling wave solutions of two different types of equations. Feng et al. [24] used the method to seek solutions of the Kolmogorov-Petrovskii-Piskunov equation. Liu et al. [25] concerned the method to simplified MCH equation and Gardner equation. Feng [26] applied the method to the seventh-order Sawada-Kotera equation.

At present time, Guo and Zhou [27] first proposed the extended -expansion method based on new ansatz. They applied the method to the Whitham-Broer-Kaup-Like equations and couple Hirota-Satsuma KdV equations. Zayed et al. [28] used extended -expansion method to some nonlinear PDEs in mathematical physics. Zhang et al. [29] presented an improved -expansion method for solving nonlinear evolution equations (NLEEs). Li et al. [30] introduced -expansion method to obtain traveling wave solutions of the Zakharov equations. Hayek [31] proposed extended -expansion method for constructing exact solutions to the KdV Burgers equations with power-law nonlinearity.

In this paper, our aim is to investigate abundant new exact traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon equation by the -expansion method.

2. Description of the -Expansion Method

Suppose that the nonlinear partial differential equation is of the form where is an unknown function and is a polynomial in which has various partial derivatives, and the highest order derivatives and nonlinear terms are involved.

The main steps of the -expansion method [21] are conveyed in the following.

Step 1. The traveling wave transformation: where is the wave speed, is the combination of two independent variables and , transform (2.1) into an ordinary differential equation for where primes denote the ordinary derivatives with respect to.

Step 2. If possible, integrate (2.3) term by term for one or more times, yields constant(s) of integration. For simplicity, the integration constant(s) may be zero.

Step 3. Suppose that the solution of (2.3) can be expressed by a polynomial in : where () are constants and , and satisfies the second order linear ordinary differential equation (LODE): where and are arbitrary constants. The positive integer can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in (2.3).

Step 4. Substituting (2.4) together with (2.5) into (2.3) yields an algebraic equation involving powers of . Then, equating coefficients of each power of to zero yields a set of algebraic equations for (), , , and . Since the general solution of (2.5) is known for us, then substituting () and together with general solution of (2.5) into (2.4), we obtain exact traveling wave solutions of the nonlinear partial differential equation (2.1).

3. Application of the Method

In this section, we apply the method to construct the hyperbolic, the trigonometric, and the rational function solutions of the fifth-order Caudrey-Dodd-Gibbon equation, and the solutions are shown in graphs.

3.1. The Caudrey-Dodd-Gibbon Equation

We consider the fifth-order Caudrey-Dodd-Gibbon equation: Now, we use the transformation equation (2.2) into (3.1), which yields, where primes denote the derivative with respect to .

Equation. (3.2) is integrable, therefore, integrating once with respect to yields where is an integral constant, that is, to be determined later.

Considering the homogeneous balance between and in (3.3), we obtain . Therefore, the solution of (3.3) is of the form: where , , and are constants and . Substituting (3.4) together with (2.5) into (3.3) and collecting all terms with the same power of and setting each coefficient of the resulted polynomial to zero, we obtain a set of algebraic equations for , , , , , , and as follows: Solving the system of algebraic equations with the aid of Maple 13, we obtain two different sets of solution.

Case 1. One has where and are arbitrary constants.

Case 2. One has where , , and are arbitrary constants.

Case 1. Substituting (3.6) into (3.4) yields

Substituting the general solution of (2.5) into (3.8), we obtain three types of traveling wave solutions of (3.3) as follows.

Hyperbolic Function SolutionsWhen , substituting the general solution of (2.5) into (3.8), we obtain the following traveling wave solution of (3.3): where , , and are arbitrary constants. The various known results can be rediscovered, if and are taken as special values. For example:

(i)if but we obtain (ii)If but , we obtain (iii)If and , we obtain where .

Trigonometric Function SolutionsWhen , we obtain where , and are arbitrary constants.

The various known results can be rediscovered, if and are taken as special values.

Rational Function SolutionWhen we obtain where , , and are arbitrary constants.

Case 2. Substituting (3.7) into (3.4) yields

Substituting the general solution of (2.5) into (3.15), we obtain three types of traveling wave solutions of (3.3) as follows.

Hyperbolic Function SolutionsWhen , we obtain where . and are arbitrary constants.

The various known results can be rediscovered, if and are taken as special values.

Trigonometric Function SolutionsWhen , we obtain where and are arbitrary constants. The various known results can be rediscovered, if and are taken as special values. For example:

(i)if but , we obtain (ii)If but we obtain (iii)If and , we obtain where .

Rational Function SolutionWhen , where . and are arbitrary constants.

3.2. Discussion

The solutions of the CDG (3.1) are investigated by different methods, such as Jin [32] investigated solutions by the variational iteration method, Salas [33] by the projective Riccati equation method, and Wazwaz [34] by using the tanh method. To the best of our awareness the CDG equation is not solved by the prominent -expansion method. In this paper, we solve this equation by the -expansion method. It is noteworthy to point out that our attained solutions are new and cannot be found from above author’s solutions by any choice of arbitrary constants.

3.3. Graphical Representations of the Solutions

The solutions are shown in the graphs with the aid of Maple 13 in Figures 1–5.

Figure 1: Solitary wave obtained from solution (3.10), for , .

Figure 2: Solitary wave obtained from solution (3.11), for , .

Figure 3: Solitary wave obtained from solution (3.14), for , , , .

Figure 4: Solitary wave obtained from solution (3.19), for , , .

Figure 5: Solitary wave obtained from solution (3.19), for , , .

4. Conclusions

In this paper, three types of traveling wave solutions, such as the hyperbolic, the trigonometric, and the rational functions of the Caudrey-Dodd-Gibbon equation are successfully obtained by using the -expansion method. Exact traveling wave solutions of this equation have many potential applications in engineering and mathematical physics. The obtained solutions also show that the method is effective, more powerful, and simple for searching exact traveling wave solutions of the NLPDEs. The method can be applied in different types of NLEEs and it is our task in the future.

Acknowledgments

This paper is supported by USM short term Grant (no. 304/PMATHS/6310072). The authors would like to thank the School of Mathematical Sciences, USM, for providing related research facilities.

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