Analysis of differential equations involving Caputo\u2013Fabrizio fractional operator and its applications to reaction\u2013diffusion equations

Amjad Shaikh,Dumitru Baleanu,Asifa Tassaddiq,Kottakkaran Sooppy Nisar

doi:10.1186/s13662-019-2115-3

Abstract

This manuscript deals with fractional differential equations including Caputo–Fabrizio differential operator. The conditions for existence and uniqueness of solutions of fractional initial value problems is established using fixed point theorem and contraction principle, respectively. As an application, the iterative Laplace transform method (ILTM) is used to get an approximate solutions for nonlinear fractional reaction–diffusion equations, namely the Fitzhugh–Nagumo equation and the Fisher equation in the Caputo–Fabrizio sense. The obtained approximate solutions are compared with other available solutions from existing methods by using graphical representations and numerical computations. The results reveal that the proposed method is most suitable in terms of computational cost efficiency, and accuracy which can be applied to find solutions of nonlinear fractional reaction–diffusion equations.

Highlights

Nowadays, the mathematical models involving fractional order derivative were given noticeable importance because they are more accurate and realistic as compared to the classical order models [22, 26, 28]
Motivated by the advancement of fractional calculus, many researchers have focused to investigate the solutions of nonlinear differential equations with the fractional operator by developing quite a few analytical or numerical techniques to find approximate solutions [6, 10, 19, 29, 30]
The residual power series method (RPSM) was applied by Tchier et al [31] to find a numerical solution of fractional reaction–diffusion equations

Summary

Introduction

The mathematical models involving fractional order derivative were given noticeable importance because they are more accurate and realistic as compared to the classical order models [22, 26, 28]. We analyze the following Caputo–Fabrizio fractional differential equations (C-FFDE) to obtain uniqueness and existence criteria of solutions [5, 7, 35]: [24] Khan et al used the homotopy analysis method (HAM) to find approximate analytical solutions of fractional reaction–diffusion equations.

Results

Conclusion