Regularization of sideways problem for a time fractional diffusion equation with nonlinear source

TL;DR

This paper addresses the ill-posed inverse problem for a time-fractional diffusion equation with a nonlinear source, proposing a new regularization method and demonstrating its effectiveness through numerical examples.

Contribution

It introduces a novel regularization approach for stabilizing the ill-posed inverse problem in fractional diffusion equations with nonlinear sources.

Findings

The inverse problem is proven to be ill-posed in the sense of Hadamard.

A new regularization method effectively stabilizes the solution.

Numerical examples validate the proposed regularization technique.

Abstract

In this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e. the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak {\color{black} a} priori assumptions on the sought solution, we propose a new regularization method for stabili{\color{black}z}ing the ill-posed problem. We also provide a numerical example to illustrate our results.