Exact Solutions to Some Nonlinear Partial Differential Equations in Mathematical Physics Via the (G'/G) -Expansion Method

M Ali Akbar,Norhashidah Hj Mohd Ali

doi:10.19026/rjaset.6.3556

Abstract

The (G􏲓′/􏲓G)-expansion method is a powerful tool for the direct analysis of contender nonlinear equations. In this study, we search new exact traveling wave solutions to some nonlinear partial differential equations, such as, the Kuramoto-Sivashinsky equation, the Kawahara equation and the Carleman equations by means of the (􏲓G􏲓′/􏲓G)- expansion method which are very significant in mathematical physics. The solutions are presented in terms of the hyperbolic and the trigonometric functions involving free parameters. It is shown that the novel (􏲓G􏲓′/􏲓G􏲓)-expansion method is a competent and influential tool in solving nonlinear partial differential equations in mathematical physics.

Highlights

Nonlinear processes are one of the biggest challenges and not easy to control because the nonlinear characteristic of the system abruptly changes due to some small changes of valid parameters including time
The studies of exact solutions of Nonlinear Partial Differential Equations (NLPDEs) play a crucial role to understand the physical mechanism of nonlinear phenomena
Our aim in this study is to present an application of the (GG′/GG)-expansion method to some nonlinear PDEs in mathematical physics namely, the KuramotoSivashinsky equation, the Kawahara equation and the Carleman equations

Summary

INTRODUCTION

Nonlinear processes are one of the biggest challenges and not easy to control because the nonlinear characteristic of the system abruptly changes due to some small changes of valid parameters including time. Very lately, Wang et al (2008) developed a new method called the (GG′/GG) -expansion method to look for travelling wave solutions of nonlinear evolution equations. By using the (GG′/GG)-expansion method, Wang et al (2008) successfully obtained travelling wave solutions of four nonlinear partial differential equations. Using the (GG′/GG)-expansion method, Zheng (2011) obtained the travelling wave solutions of the sixth-order Drinfeld-Sokolov-Satsuma-Hirota equation. Akbar et al (2012a) obtained abundant traveling wave solutions of the Generalized Bretherton equation by employing the (GG′/GG)-expansion method. Akbar et al (2012b) obtained more new exact solutions of some NLPDEs by a generalized and improved (GG′/GG) -expansion method. Substantial work has to be done in order for the (GG′/GG)-expansion method to be well established, since every nonlinear equation has its own physically significant rich structure. Our aim in this study is to present an application of the (GG′/GG)-expansion method to some nonlinear PDEs in mathematical physics namely, the KuramotoSivashinsky equation, the Kawahara equation and the Carleman equations

METHODOLOGY

CONCLUSION