Identifying a fractional order and a time-dependent coefficient in a time-fractional diffusion wave equation

Abstract

In this article, we study a nonlinear inverse problem of identifying a fractional order of derivative and a time-dependent reaction coefficient in a time-fractional diffusion wave equation from integral data. We prove the uniqueness of determining the fractional order and the time-dependent reaction coefficient simultaneously. In addition, we provide a Bayesian method to reconstruct numerically the time-dependent reaction coefficient and the fractional order. By the continuity of the forward mapping with respect to the unknown parameter, we show the well-definedness and well-posedness of the posterior measure induced by the Bayesian method for the inverse time coefficient problem. Moreover, we adopt an iterative regularizing ensemble Kalman method to solve the Bayesian inverse problem and present some numerical examples to confirm the effectiveness of the proposed method.