An Efficient Method for Solving Coupled Time Fractional Nonlinear Evolution Equations with Conformable Fractional Derivatives

Osama H Mohammed,Firas S Ahmed

doi:10.24996/ijs.2020.61.11.29

Osama H Mohammed, Firas S Ahmed

Open Access

https://doi.org/10.24996/ijs.2020.61.11.29

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Abstract

In this article, an efficient reliable method, which is the residual power series method (RPSM), is used in order to investigate the approximate solutions of conformable time fractional nonlinear evolution equations with conformable derivatives under initial conditions. In particular, two types of equations are considered, which are time coupled diffusion-reaction equations (CD-REs) and MKdv equations coupled with conformable fractional time derivative of order Î±. The attitude of RPSM and the influence of different values of Î± are shown graphically.

Highlights

The subject of fractional calculus has been gaining a considerable attention from various authors due to its important role in many applications
Khalil et al.[8, 9] were the first who proposed a new fractional derivative, viz. the conformable fractional derivative (CFD) to take control of the remarkable problem that had occurred in the Riemann-Liouville and Caputo fractional derivatives, which is the inheritance of the nonlocal properties from the integral
The residual power series method (RPSM) suggests the solution for Eqs.(4) as a fractional power series (FPS) about the (IC)t=0 as:

Summary

Introduction

The subject of fractional calculus has been gaining a considerable attention from various authors due to its important role in many applications. There are many definitions for the fractional order derivatives, such as those reported by Riemann-Liouville, Caputo, and Grunwald-Letnikov, etc.[5,6,7]. RPSM was used to investigate a numerical solution for the fractional Burger equation[15]. The exact analytical solution of the time-fractional Schrodinger equation was found in another work [16]. The main aim of this paper is to employ RPSM for two models of nonlinear FDEs of special interest physically, in terms of the convergent fractional power series. 2. Preliminaries Definition 2.1[8]: Given a function y :[0, ) , the conformable fractional derivative of order α of y is defined by CDt The RPSM suggests the solution for Eqs.(4) as a fractional power series (FPS) about the (IC)t=0 as:. Finding f 1(x ), f 2 (x ), f 3(x ), needs to solve the algebraic equations: C

RPSM for solving CD-REs

Conclusions