Inverse $t$-source problem and a strict positivity property for coupled subdiffusion systems

TL;DR

This paper addresses the inverse problem of identifying the temporal source in coupled fractional diffusion systems using single-point observations, establishing stability, uniqueness, and proposing a practical iterative numerical method.

Contribution

It introduces a novel stability and uniqueness analysis for the inverse source problem and develops an effective ensemble Kalman method for numerical reconstruction.

Findings

Lipschitz stability under non-degeneracy conditions

Strict positivity of fractional integrals ensures uniqueness

Numerical method demonstrates high accuracy and robustness

Abstract

This article is concerned with the inverse problem on determining the temporal component of the source term in a coupled system of time-fractional diffusion equations by single point observation. Under a non-degeneracy condition on the known spatial component, we establish the Lipschitz stability by observing all solution components by a series representation of the mild solution. To reduce the observation data, we prove the strict positivity of some fractional integral of the solution to the homogeneous problem by a modified Picard iteration. This, together with a coupled Duhamel's principle, lead us to the uniqueness of the inverse problem by observing any single solution component under a specific structural constraint on the unknown. Numerically, we propose an iterative regularizing ensemble Kalman method (IREKM) for the simultaneous recovery of the temporal sources. Through extensive numerical tests, we demonstrate its accuracy, robustness against noise, and scalability with respect to the number of components. Our findings highlight the essential roles of the non-degeneracy condition, measurement configuration, and fractional structural constraints in ensuring reliable reconstructions. The proposed framework provides both rigorous theoretical guarantees and a practical algorithmic approach for multi-component source identification in fractional diffusion systems.