Modelling, analysis, and numerical methods for a geometric inverse source problem in variable-order time-fractional subdiffusion

Abstract

There exist research works on studying time-dependent integer-order and time-fractional constant-order geometric inverse source problems in the literature. The time-fractional variable-order geometric inverse source problems although also have important physical applications have not been studied mathematically and numerically in literature. The aim of this work is to study an inverse source problem associated with a variable-order time-fractional subdiffusion equation. We first build a mathematical model and show existence of the optimal shape for shape reconstruction of the source support. Then, shape sensitivity analysis is performed to propose a shape gradient optimization algorithm allowing deformations for numerically solving the model problem. In order to reconstruct the source support with topology unknown a priori, moreover, we build a phase-field model and propose a gradient algorithm allowing both shape and topological changes by a phase-field method. A variety of numerical examples are presented to demonstrate effectiveness of the two algorithms.