Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel

Mohammed Al-Refai,Thabet Abdeljawad

doi:10.1186/s13662-017-1356-2

Abstract

In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

Highlights

Fractional diffusion models (FDM) are generalization to the diffusion models with integer derivatives
6 Concluding remarks We have considered linear and nonlinear fractional diffusion equations with Caputo fractional derivative of non-singular kernel
We have obtained an estimate of the Caputo fractional derivative of non-singular kernel of a function at its extreme points

Summary

Introduction

Fractional diffusion models (FDM) are generalization to the diffusion models with integer derivatives. Existence and uniqueness results were established by the new maximum principles obtained by estimating the fractional derivative of a function at its extreme points. The applicability of maximum principles for the linear and nonlinear fractional diffusion systems with the Riemann-Liouville fractional derivative was discussed and proved for the first time by Al-Refai and Luchko in [ ], where existence, uniqueness and stability results were established. We extend the results presented in [ ] for the fractional diffusion equations with the Caputo fractional derivative of non-singular kernel. In Section , an estimate of the Caputo fractional derivative of non-singular kernel of a function at its extreme points is deduced in a form of certain inequality.

The associated fractional integral is defined by CF Iaα f

The last inequality yields

Applying the simple maximum principle will lead to