Simultaneous Determination of the Order and a Coefficient in a Fractional Diffusion-Wave Equation

This paper recovers the order of fractional derivative and a time-dependent potential coefficient in a time-fractional diffusion wave equation by an integral condition or one single point measurement on the boundary. The Lipschitz continuity of the forward operators from the unknown order and coefficient to the given data are achieved in terms of the integral equation held by the solution of the direct problem. We also obtain the uniqueness for the considered inverse problems in terms of somewhat general conditions to the given functions. Moreover, we propose a Tikhonov-type regularization method and prove the existence of the regularized solution and its convergence to the exact solution under a suitable regularization parameter choice. Then we use a linearized iteration algorithm to recover numerically the order and time-dependent potential coefficient simultaneously. Three numerical examples for one- and two-dimensional cases are provided to display the efficient of the proposed method.