On Fractional Diffusion Equation with Caputo-Fabrizio Derivative and Memory Term

Binh Duy Ho,Nguyen Hoang Luc,Mustafa Inc,Long Le Dinh,Van Kim Ho Thi,Phuong Nguyen

doi:10.1155/2021/9259967

Abstract

In this paper, we examine a nonlinear fractional diffusion equation containing viscosity terms with derivative in the sense of Caputo-Fabrizio. First, we establish the local existence and uniqueness of lightweight solutions under some assumptions about the input data. Then, we get the global solution using some new techniques. Our main idea is to combine theories of Banach’s fixed point theorem, Hilbert scale theory of space, and some Sobolev embedding.

Highlights

The fractional calculation has a long history and plays an important role in the simulation of physical phenomena or real life, for example, mechanics, electricity, chemistry, biology, economics, notably control theory, and images
In a series of research directions on fractional differential equations (FDE), the most prominent is the appearance of two derivatives: Caputo derivative and Riemann-Liouville derivative
The difficulty in studying this problem is from the memory viscoelastic model appearing in the main equation. This term makes some of the assessments more complicated. Another difficulty is the study of the existence of global solutions

Summary

Introduction

The fractional calculation has a long history and plays an important role in the simulation of physical phenomena or real life, for example, mechanics, electricity, chemistry, biology, economics, notably control theory, and images. The Caputo-Fabrizio fractional derivative was presented in 2015 [25] with the aim of avoiding singular kernels. It is the convolution of the exponential function and the first-order derivative. Our main aim in this paper is to provide the local and global existence for problem (1) under some various assumptions on the input data. The difficulty in studying this problem is from the memory viscoelastic model appearing in the main equation. This term makes some of the assessments more complicated. Another difficulty is the study of the existence of global solutions.

Preliminaries

Local Existence Results

Global Existence Results under a Global Lipschitz Case

Conclusion