The expa function method and the conformable time-fractional KdV equations

Asim Zafar

doi:10.1515/nleng-2018-0094

Abstract

Abstract The nonlinear fractional differential equations (FDEs) are produced by mathematical modelling of some nonlinear physical systems. The study of such nonlinear physical models through wave solutions analysis corresponding to their FDEs, has a dynamic role in applied sciences. In this paper, we are going to explore the conformable time-fractional KdV equations using the expa function method. The way to reach explicit exact wave solutions is to transform the fractional order PDE into a nonlinear ODE of discrete order through travelling wave transforms. The subsequent equation has been explored by utilizing the exp a function approach. Consequently, some new explicit exact wave solutions of the said equations are effectively formulated and graphically conveyed with the help of numerical simulation.

Highlights

The nonlinear fractional di erential equations (FDEs) are produced by mathematical modelling of some nonlinear physical systems. The study of such nonlinear physical models through wave solutions analysis corresponding to their FDEs, has a dynamic role in applied sciences
The Korteweg– de Vries (KdV) equations have been observed in a broad variety of material science phenomena as a model for the development and communication of nonlinear waves
The KdV equation was introduced by Korteweg and de Vries to describe shallow water waves of long wavelength with small amplitude

Summary

Introduction

The Korteweg– de Vries (KdV) equations have been observed in a broad variety of material science phenomena as a model for the development and communication of nonlinear waves. The explicit exact wave solutions have always been a particular importance among the researchers in many elds of nonlinear sciences. Various signi cant methods have been presented to explore exact solutions in di erent research articles, for example, the ansatz method, modied simple equation method, the extended trail equation, the rst integral, the improved tan(φ(η)/ )-expansion, the exp-function and the exp(−φ(ε)) methods [15,16,17,18,19,20,21]. The paper’s aim is to apply the expa function approach to nonlinear conformable time-fractional KdV equations.

This work is licensed under the Creative Commons Attribution alone

Explicit exact wave solutions for time fractional KdV equations

Conclusion