On the simultaneous recovery of the fractional order and a time-dependent source term in a time-fractional diffusion-wave equation

We consider the simultaneous recovery of the fractional order and a time-dependent source term in a time-fractional diffusion-wave equation. This gives rise to a coupled nonlinear inverse problem, in which the two unknowns are intrinsically linked through the nonlocal dynamics of the fractional model. We first establish a uniqueness result for the joint identification of the fractional order and the source term, thereby providing a rigorous theoretical foundation for the inversion. Owing to the severe ill-posedness of the problem, direct reconstruction from noisy data is highly unstable. To address this difficulty, we adopt a Bayesian formulation, which allows prior information and observational uncertainty to be incorporated systematically. Based on this framework, we develop an iterative regularized ensemble Kalman method for the resulting statistical inverse problem. The proposed approach combines the computational efficiency of ensemble-based methods with the stabilizing effect of iterative regularization, making it well suited to this class of nonlinear inverse problems. Numerical experiments show that the method yields accurate and stable reconstructions of both the fractional order and the time-dependent source term, even in the presence of measurement noise.