Abstract
<title>Abstract</title> <p>This paper is devoted to the investigation of an inverse source problem associated with a fractional time-space diffusion equation. The main goal is to recover a space-dependent source term from noisy final-time observations. First, the well-posedness of the corresponding direct problem is established through a rigorous theoretical analysis. Subsequently, the inverse problem, which is inherently ill-posed, is reformulated as an optimization problem. In this framework, we study the existence of minimizers together with the uniqueness and stability properties of the proposed formulation. To overcome the ill-posedness, the Tikhonov regularization technique is employed, and the resulting regularized solution is computed using a gradient descent algorithm. Finally, several numerical reconstruction experiments are presented to illustrate the effectiveness, stability, and accuracy of the proposed approach.</p>