Solutions of the time fractional reaction–diffusion equations with residual power series method

Fairouz Tchier,Zeliha S Korpinar,Dumitru Baleanu,Mustafa Inc

doi:10.1177/1687814016670867

Abstract

In this article, the residual power series method for solving nonlinear time fractional reaction–diffusion equations is introduced. Residual power series algorithm gets Maclaurin expansion of the solution. The algorithm is tested on Fitzhugh–Nagumo and generalized Fisher equations with nonlinearity ranging. The solutions of our equation are computed in the form of rapidly convergent series with easily calculable components using Mathematica software package. Reliability of the method is given by graphical consequences, and series solutions are used to illustrate the solution. The found consequences show that the method is a powerful and efficient method in determination of solution of the time fractional reaction–diffusion equations.

Highlights

In the last few years, there has been considerable interest in fractional calculus used in many fields, such as regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, viscoelasticity, electrical circuits, electro-analytical chemistry, biology, and control theory.[1,2,3,4] Besides, there has been a significant theoretical development in fractional differential equations and its applications.[5,6,7,8,9,10] fractional derivatives supply an important implement for the definition of hereditary characteristics of different necessaries and treatments
It has been successfully put into practice to handle the approximate solution of the generalized Lane-Emden equation,[12] the solution of composite and non-composite fractional differential equations,[13] predicting and representing the multiplicity of solutions to boundary value problems of fractional order,[14] constructing and predicting the solitary pattern solutions for nonlinear time fractional
The fundamental objective of this article is to introduce an algorithmic form and implement a new analytical repeated algorithm derived from the residual power series (RPS) to find numerical solutions for the time fractional reaction– diffusion equation

Summary

Introduction

In the last few years, there has been considerable interest in fractional calculus used in many fields, such as regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, viscoelasticity, electrical circuits, electro-analytical chemistry, biology, and control theory.[1,2,3,4] Besides, there has been a significant theoretical development in fractional differential equations and its applications.[5,6,7,8,9,10] fractional derivatives supply an important implement for the definition of hereditary characteristics of different necessaries and treatments. In section ‘‘Applications for RPSM algorithm and graphical results,’’ the base opinion of the RPSM is constituted to construct the solution of the time fractional nonlinear reaction–diffusion equations and some graphical consequences are included to demonstrate the reliability and efficiency of the method. For n to be the smallest integer that exceeds a, the Caputo time fractional derivative operator of order a of u(x, t) is defined as[13,16]

Result

Conclusion