Point Source Identification in Subdiffusion from A Posteriori Internal Measurement

TL;DR

This paper addresses the inverse problem of identifying point sources and their strengths in a subdiffusion model with fractional time derivatives, establishing uniqueness and demonstrating numerical reconstruction feasibility.

Contribution

It extends point source identification methods to subdiffusion models with fractional derivatives, proving uniqueness and providing numerical validation.

Findings

Uniqueness of source location and strength recovery in subdiffusion models.

Extension of source identification to models with time-dependent coefficients.

Numerical experiments confirming the feasibility of reconstruction.

Abstract

In this work we investigate an inverse problem of recovering point sources and their time-dependent strengths from {a posteriori} partial internal measurements in a subdiffusion model which involves a Caputo fractional derivative in time and a general second-order elliptic operator in space. We establish the well-posedness of the direct problem in the sense of transposition and improved local regularity. Using classical unique continuation of the subdiffusion model and improved local solution regularity, we prove the uniqueness of simultaneously recovering the locations of point sources, time-dependent strengths and initial condition for both one- and multi-dimensional cases. Moreover, in the one-dimensional case, the elliptic operator can have time-dependent coefficients. These results extend existing studies on point source identification for parabolic type problems. Additionally we present several numerical experiments to show the feasibility of numerical reconstruction.