Recovering the space source term for the fractional-diffusion equation with Caputo\u2013Fabrizio derivative

Le Nhat Huynh,Dumitru Baleanu,Nguyen Hoang Luc,Le Dinh Long

doi:10.1186/s13660-021-02557-3

Abstract

This article is devoted to the study of the source function for the Caputo–Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method.

Highlights

In recent times, the fractional calculus has been extensively studied
In the case the kernel has no singularities, Caputo and Fabrizio have introduced a new definition of the fractional derivatives [1]
A small error between the given observation Hε and the exact data Hε leads to a large error in the solution in output data. This is a reason why we propose some regularization methods in order to recover stable approximations for the unknown space source function

Summary

Introduction

The fractional calculus has been extensively studied. There are many definitions for the fractional derivative operator. We restore the space source term problem for a fractional diffusion equation with a Caputo–Fabrizio derivative. A small error between the given observation Hε and the exact data Hε leads to a large error in the solution in output data This is a reason why we propose some regularization methods in order to recover stable approximations for the unknown space source function. We apply the fractional Landweber method and modified quasi-boundary value method to restore the unknown space source function F. We can compare which method is the more optimal for the space source term problem for the fractional diffusion equation with the Caputo–Fabrizio derivative. Applying the fractional Landweber method given by [26], we propose the regularized solution with exact data H as follows: Fn,χ (x).

Numerical test

Conclusions