Reconstruction of a Space-Time Dependent Source in Subdiffusion Models via a Perturbation Approach

TL;DR

This paper addresses the inverse problem of reconstructing a space-time dependent source in a subdiffusion model with fractional time derivatives, establishing stability, and proposing an algorithm with numerical validation.

Contribution

It introduces a novel perturbation approach and refined regularity estimates for stable reconstruction of sources in subdiffusion models with time-dependent coefficients.

Findings

Established well-posedness and stability for the inverse problem.

Developed an efficient algorithm for source reconstruction.

Numerical results demonstrate the method's feasibility.

Abstract

In this article we study inverse problems of recovering a space-time dependent source component from the lateral boundary observation in a subidffusion model. The mathematical model involves a Djrbashian-Caputo fractional derivative of order $\alpha\in(0,1)$ in time, and a second-order elliptic operator with time-dependent coefficients. We establish a well-posedness and a conditional stability result for the inverse problems using a novel perturbation argument and refined regularity estimates of the associated direct problem. Further, we present an algorithm for efficiently and accurately reconstructing the source component, and provide several two-dimensional numerical results showing the feasibility of the recovery.