Application of fractional sub-equation method to nonlinear evolution equations

Mohamed A Abdelkawy,Anjan Biswas,Omar H El-Kalaawy,Rasha B Al-Denari

doi:10.15388/na.2018.5.5

Abstract

In this paper, we constructed a traveling wave solutions expressed by three types of functions, which are hyperbolic, trigonometric, and rational functions. By using a fractional sub-equation method for some space-time fractional nonlinear partial differential equations (FNPDE), which are considered models for different phenomena in natural and social sciences fields like engineering, physics, geology, etc. This method is a very effective and easy to investigate exact traveling wave solutions to FNPDE with the aid of the modified Riemann–Liouville derivative.

Highlights

Over the last decades, the field of fractional calculus has thrived in pure mathematics as well as in scientific applications, and its utility has become more and more conspicuous
We use a fractional subequation method to introduce another solutions for the mentioned problems in the sense of the modified Riemann–Liouville derivative defined by Jumarie [16, 17], which is a fractional version of the known (G /G) method
We will construct solutions for some nonlinear fractional nonlinear partial differential equations (FNPDE), namely the spacetime fractional Cahn–Hilliard equation, the space-time fractional fifth-order Sawda– Kotera equation, and the space-time fractional modified equal-width equation by applying the fractional subequation method in which FNPDEs are very important in mathematical physics and have been paid attention by many researchers

Summary

Introduction

The field of fractional calculus has thrived in pure mathematics as well as in scientific applications, and its utility has become more and more conspicuous. Time-fractional diffusion equation is solved using spectral decomposition method with Fourier and Laplace transforms [3]. We use a fractional subequation method to introduce another solutions for the mentioned problems in the sense of the modified Riemann–Liouville derivative defined by Jumarie [16, 17], which is a fractional version of the known (G /G) method. This method is based on the fractional ODE.

Description of a fractional subequation method

Applications

The space-time fractional Cahn–Hilliard equation

The space-time fractional fifth-order Sawda–Kotera equation

Conclusion