Solving a Time-Dependent Source Problem for a Time-Fractional Diffusion Equation with Two Numerical Approaches

The aim of this study is to determine a time-dependent source problem in the time-fractional diffusion equation from the additional measurement data at an inner point. By transforming the inverse problem into an minimization problem, we employ two distinct alternative approaches to solve the latter control problem. To assess the effectiveness of these approaches, we test them under conditions with and without noise. The results obtained are very interesting and encouraging, indicating the potential success of the proposed methods.