Numerical schemes for reconstructing profiles of moving sources in (time-fractional) evolution equations

TL;DR

This paper develops numerical reconstruction schemes for inverse moving source problems in (time-fractional) evolution equations, combining regularization, iterative thresholding, and elliptic methods to recover source profiles.

Contribution

It introduces novel numerical algorithms for reconstructing moving source profiles, extending theoretical uniqueness results with practical iterative and elliptic schemes.

Findings

Effective iterative thresholding schemes for source recovery

Elliptic approach for dual-profile convection problems

Enhanced numerical methods for inverse source problems

Abstract

This article is concerned with the derivation of numerical reconstruction schemes for the inverse moving source problem on determining source profiles in (time-fractional) evolution equations. As a continuation of the theoretical result on the uniqueness, we adopt a minimization procedure with regularization to construct iterative thresholding schemes for the reduced backward problems on recovering one or two unknown initial value(s). Moreover, an elliptic approach is proposed to solve a convection equation in the case of two profiles.