An optimal filtering method for a time-fractional inverse advection-dispersion problem

We consider an inverse problem for a time-fractional advection-dispersion equation, where the measured data is given at x = 1 and the solution is sought in the interval 0 <= x < 1. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order alpha epsilon (0, 1). We show that the inverse problem for a time-fractional advection-dispersion equation is severely ill-posed and we further apply an optimal filtering regularization method to solve it, based on the solution in the frequency domain. The corresponding convergence estimates are provided. To illustrate the results, an example is constructed to show the feasibility and efficiency of the proposed method.