Integral equation methods for inversion source problem of time-space fractional diffusion-wave equation

The time-dependent source term in a time-space fractional diffusion-wave equation is identified by using the additional measurement data at an inner point. Firstly, the existence, uniqueness, and stability estimates of the direct problem solution are proven, as well as the uniqueness and stability of the inverse source problem. Then, the inverse source problem is transformed into an equivalent first-kind Volterra integral equation. The boundary element method is employed for discretization, and the generalized Tikhonov regularization method is applied to solve the integral equation. The regularization parameter is selected using the generalized cross-validation (GCV) method to obtain a stable numerical approximation for the inverse source problem. Finally, the effectiveness and stability of the proposed method are demonstrated through five one-dimensional and two-dimensional numerical examples.