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

This paper is devoted to solve an inverse time-dependent source problem for a time-fractional diffusion-wave equation. The source function in the time-fractional diffusion-wave equation is assumed as a multiplication of a temporal function and a spatial function. We try to identify the time-dependent source term according to an additional solute concentration distribution measured at a point on the boundary or an inner point in the solution domain. Firstly, we discuss the continuity of the weak solution for the direct problem. Then, we transform the inverse problem into a first kind of Volterra integral equation and show its ill-posedness. We use a generalized Tikhonov regularization to solve the Volterra integral equation. The generalized cross-validation choice rule is applied to find a suitable regularization parameter. Lastly, we test three examples for the inverse time-dependent source problem and show the effectiveness of the proposed regularization method.