Reconstruction of a time-dependent source term in a time-fractional diffusion equation

This paper is devoted to determine a time-dependent source term in a time-fractional diffusion equation by using the usual initial and boundary data and an additional measurement data at an inner point. Based on the separation of variables and Duhamelʹs principle, we transform the inverse source problem into a first kind Volterra integral equation with the source term as the unknown function and then show the ill-posedness of the problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the Volterra integral equation of the fist kind. The generalized cross-validation choice rule is applied to find a suitable regularization parameter. Four numerical examples are provided to show the effectiveness and robustness of the proposed method.