A control-based spatial source reconstruction in fractional heat equations

This article addresses the inverse source problem for a nonlocal heat equation involving the fractional Laplacian. The primary goal is to reconstruct the spatial component of the source term from partial observations of the system's state and its time derivative over a subset of the domain. A reconstruction formula for the Fourier coefficients of the unknown source is derived, leveraging the null controllability property of the fractional heat equation when the fractional order lies in the interval $s\in(1/2,1)$. The methodology builds on spectral analysis and Volterra integral equations, providing a robust framework for recovering spatial sources under limited measurement data. Numerical experiments confirm the accuracy and stability of the proposed approach.