Inverse source problems for identifying time and space-dependent coefficients in a 2D generalized diffusion equation

This article addresses two inverse source problems related to determining a space-dependent source term and a time-dependent coefficient in a two-dimensional generalized diffusion equation. The considered problems are ill-posed in the Hadamard sense, where small perturbations in the data can lead to uncontrolled variations in the solution. The present work also provides existence and uniqueness results for the solutions of these problems under appropriate over-specified and regularity conditions. Special cases of the diffusion equation are examined, focusing on specific selections of the memory kernel involved in the time-fractional derivative. The results are illustrated with several examples, demonstrating the practical implications of the proposed methods for inverse source problems.