Reconstruction of a Space-Time-Dependent Source in Subdiffusion Models via a Perturbation Approach
Bangti Jin, Yavar Kian, Zhi Zhou
Abstract
In this article we study two inverse problems of recovering a space-time-dependent source component from the lateral boundary observation in a subdiffusion 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 a numerical algorithm for efficiently and accurately reconstructing the source component, and we provide several two-dimensional numerical results showing the feasibility of the recovery.