Identifying the order and a time-dependent source term in a time-fractional diffusion-wave equation by a single measurement

A nonlinear inverse problem for a time-fractional diffusion-wave equation is investigated, in which both the fractional order α and the time-dependent source term q(t) are to be recovered by a single-point additional measurement data. We prove the uniqueness of the inverse problem and establish the Lipschitz continuity of the forward operator based on stability estimates of the solution. To cope with the ill-posedness of the nonlinear inverse problem, we adopt a Tikhonov-type regularization method incorporating an H1-penalty and investigate both the existence and the convergence of the corresponding regularized solution. To solve this nonlinear inverse problem efficiently, we propose an alternating minimization method integrated with an accelerated linearized iterative algorithm (AM-ALIA). The method is combined with a piecewise linear finite element approximation to simultaneously recover the approximate order and time source term. Furthermore, the numerical experiments in one-dimensional and two-dimensional settings demonstrate the robustness, accuracy, and computational efficiency of the proposed method.