The(G'/G,1/G)-Expansion Method and Its Applications for Solving Two Higher Order Nonlinear Evolution Equations

E. M. E. Zayed,K. A. E. Alurrfi

doi:10.1155/2014/746538

Abstract

The two variable(G'/G,1/G)-expansion method is employed to construct exact traveling wave solutions with parameters of two higher order nonlinear evolution equations, namely, the nonlinear Klein-Gordon equations and the nonlinear Pochhammer-Chree equations. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations are rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original(G'/G)-expansion method proposed by Wang et al. It is shown that the two variable(G'/G,1/G)-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.

Highlights

Investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena
The key idea of the one variable (G󸀠/G)-expansion method is that the exact solutions of nonlinear PDEs can be expressed by a polynomial in one variable (G󸀠/G) in which G = G(ξ) satisfies the second order linear ODE G󸀠󸀠(ξ) + λG󸀠(ξ) + μG(ξ) = 0, where λ and μ are constants and 󸀠 = d/dξ
The key idea of the two variable (G󸀠/G, 1/G)-expansion method is that the exact traveling wave solutions of nonlinear PDEs can be expressed by a polynomial in two variables (G󸀠/G) and (1/G) in which G = G(ξ) satisfies the second order linear ODE G󸀠󸀠(ξ) + λG(ξ) = μ, where λ and μ are constants

Summary

Introduction

Investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena. The key idea of the one variable (G󸀠/G)-expansion method is that the exact solutions of nonlinear PDEs can be expressed by a polynomial in one variable (G󸀠/G) in which G = G(ξ) satisfies the second order linear ODE G󸀠󸀠(ξ) + λG󸀠(ξ) + μG(ξ) = 0, where λ and μ are constants and 󸀠 = d/dξ. The key idea of the two variable (G󸀠/G, 1/G)-expansion method is that the exact traveling wave solutions of nonlinear PDEs can be expressed by a polynomial in two variables (G󸀠/G) and (1/G) in which G = G(ξ) satisfies the second order linear ODE G󸀠󸀠(ξ) + λG(ξ) = μ, where λ and μ are constants. The objective of this paper is to apply the two variable (G󸀠/G, 1/G)-expansion method to find the exact traveling wave solutions of the higher order nonlinear Klein-Gordon equations [35].

Applications

Result

Conclusions