A direct sampling method for time-fractional diffusion equation

This paper introduces a direct sampling method tailored for identifying the location of the source term within a time-fractional diffusion equation (TFDE). The key aspect of our approach involves the utilization of a versatile family of index functions, which can be chosen according to the specific characteristics of the source term. Recognizing the key role of the TFDE's fundamental solution within the index function, we further enhance our method by deriving its asymptotic expansions. This advancement not only enhances the accuracy, but also significantly improves the computational efficiency of our method. To validate the effectiveness and robustness of the proposed sampling method, we conduct a series of comprehensive numerical experiments.