Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations

TL;DR

This paper establishes a weak unique continuation property for time-fractional diffusion-advection equations, enabling the unique determination of source terms from interior measurements and proposing an efficient numerical solution method.

Contribution

It introduces a novel weak unique continuation property for time-fractional equations and applies it to solve an inverse source problem with a new iterative algorithm.

Findings

Proved weak unique continuation property using Laplace transform.

Established uniqueness in inverse source problem from interior data.

Demonstrated the efficiency of the proposed numerical algorithm.

Abstract

In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iteration thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.