Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2-Expansion Method

Sekson Sirisubtawee,Sanoe Koonprasert

doi:10.1155/2018/7628651

Abstract

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

Highlights

Various phenomena such as shallow water waves and multicellular biological dynamics arising in the nonlinear physical sciences [1, 2], engineering [3, 4], and biology [5] can be modeled by a class of integrable nonlinear evolution equations which can be expressed in terms of nonlinear partial differential equations (NPDEs) of integer orders
Study of traveling wave solutions of NPDEs plays a significant role in the investigation of behaviors of nonlinear phenomena
The (G󸀠/G2)-expansion method has been applied to find some new forms of the explicit exact solutions of the three problems, i.e., the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system

Summary

Introduction

Various phenomena such as shallow water waves and multicellular biological dynamics arising in the nonlinear physical sciences [1, 2], engineering [3, 4], and biology [5] can be modeled by a class of integrable nonlinear evolution equations which can be expressed in terms of nonlinear partial differential equations (NPDEs) of integer orders. Arnous [36] investigated the use of the modified (w/g)expansion method for finding traveling wave solutions of nonlinear evolution equations. Gepreel [39] employed the extended rational (G󸀠/G2)-expansion method to obtain traveling wave solutions of the first equation of two integral members of nonlinear Kadomtsev-Petviashvili (KP) hierarchy equations in mathematical physics. Mohyud-Din and Bibi [40] used the (G󸀠/G2)-expansion method along with the fractional complex transform to analytically solve the space-time fractional Zakharov-Kuznetsov-Benjamin-BonaMahony (ZKBBM) and the space-time fractional coupled Burgers equations for innovative exact solutions. ±M, where M is some positive integer) and setting all of the obtained coefficients to zero, we acquire a system of nonlinear algebraic equations for the unknown constants a0, aj, bj

C C cosh cosh

Result

Conclusions