Abstract and Applied Analysis

Volume 2014, Article ID 252168, 4 pages

http://dx.doi.org/10.1155/2014/252168

## Exact Solutions for Nonlinear Wave Equations by the Exp-Function Method

^{1}College of Information Science & Technology, Zhengzhou Normal University, Zhengzhou 450044, China^{2}Institute of Systems Engineering, Huazhong University of Science & Technology, Wuhan 430074, China^{3}College of Computer, Wuhan University of Technology, Wuhan 430070, China

Received 14 March 2014; Revised 20 April 2014; Accepted 21 April 2014; Published 6 May 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2014 Mingsheng Hu 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

This paper elucidates the main advantages of the exp-function method in finding exact solutions of nonlinear wave equations. By the aid of some mathematical software, the solution process becomes extremely simple and accessible.

#### 1. Introduction

One of the most important aspects in nonlinear science is how to solve an exact solution of a nonlinear equation. Recently many different methods have appeared, among which the homotopy perturbation method [1–4], the tanh-method [5], the sinh-method [6, 7], and the F-expansion method [8–11] have caught much attention; however, all these methods are valid for some special kinds of nonlinear equations. It is therefore very much needed to find a universal approach to nonlinear equations; this is very challenging indeed, and the exp-function method [12–15] meets this requirement. The exp-function method itself is mathematically beautiful and extremely accessible to nonmathematicians. The use of the method requires no special knowledge of advanced calculus, and it is especially effective for solitary solutions.

#### 2. Exp-Function Method

The exp-function method was first proposed by He and Wu [16], and we consider a general partial differential equation (PED) in the form to pick out the main solution process and its advantages.

Use a transformation [16] where , and are unknown constants and should be determined later. By (2), we can convert (1) to the following nonlinear ordinary differential equation: According to the exp-function method, we assume that its solution can be expressed in the following form [16, 17]: where are positive integers that could be freely chosen. To determine the value of and , we balance the linear term of highest order of (3) with the highest order of the nonlinear term. Similarly for determining the value of and , we balance the lowest orders of linear and nonlinear terms in (3). By substituting (4) into (3), collecting terms of the same term of , and equating the coefficient of each power of exp to zero, we can get a set of algebraic equations for determining unknown constants.

#### 3. Exact Solution for Nonlinear Wave Equation

In order to illustrate the basic solution process of the exp-function method, we use the Burgers-Huxley equation as an example, which can be expressed as [18] where is an unknown function, are the partial derivatives of with respect to and , respectively, and and are arbitrary constants.

According to the exp-function method [15–17], we introduce a complex variation defined as Equation (5) thus becomes an ordinary differential equation as We suppose that the solution of (7) can be expressed as Thus we have Balancing highest order of exp-function in (9), we have , which leads to the result . Similarly we balance the lowest orders of linear and nonlinear terms in (5) to determine values of and , and we can get . For simplicity, we set and ; then (8) reduces to Substituting (10) in to (5), we have where , Setting the coefficients of to zero, we have With the help of some mathematical software, we can solve the solutions of the algebraic equations.

*Case 1. *Consider
This implies the following exact solution:
where
are parameters, , and is a free real number.

*Case 2. *Consider
This case gives another exact solution as follows:
where , are nonzero free parameters.

*Case 3. *Consider
This results in the following exact solution:
where is nonzero free parameter.

#### 4. Conclusion

By some mathematical software, the solution process is extremely simple and abundant solutions are predicted [19–21]. The exp-function method is a universal tool for nonlinear equations and can be easily extended to fractional calculus [22–27].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This project is sponsored by the National Natural Science Foundation of China (NSFC, Grants U1204703 and U1304614), the Key Scientific and Technological Project of Henan Province (122102310004), the Fundamental Research Funds for the Central Universities (HUST: 2012QN087 and 2012QN088), and the Innovation Scientists and Technicians Troop Construction Projects of Zhengzhou City (10LJRC190).

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