Regularization operators for identifying the unknown source in the time-fractional convection-diffusion-reaction equation

TL;DR

This paper develops regularization methods to stabilize the inverse problem of identifying a time-dependent source in a time-fractional convection-diffusion-reaction equation, supported by analytical solutions and numerical examples.

Contribution

It introduces three new regularization operators and a novel parameter selection rule for the inverse source identification problem.

Findings

Proposed regularization operators improve stability of source identification.

Analytical solutions demonstrate the instability of the inverse problem.

Numerical examples validate the effectiveness of the regularization strategies.

Abstract

This article presents a mathematical study of the problem of identifying a time-dependent source term in transport processes described by a timefractional parabolic equation, based on noisy time-dependent measurements taken at an arbitrary position. The problem is analytically solved using Fourier techniques, and it is shown that the solution is unstable. To address this instability, three one-parameter families of regularization operators are proposed, each designed to counteract the factors responsible for the instability of the inverse operator. Additionally, a new rule for selecting the regularization parameter is introduced, and an error bound is derived for each estimate. Numerical examples with varying characteristics are provided to illustrate the advantages of the proposed strategies.