A New Regularization Method for the Time Fractional Inverse Advection-Dispersion Problem

Abstract

In this paper, we consider the time fractional inverse advection-dispersion problem (TFIADP) in a quarter plane. The solute concentration and dispersion flux are sought from a measured concentration history at a fixed location inside the body. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative with the Caputo fractional derivative of order $\alpha$ ($0<\alpha<1$). We show that the TFIADP is severely ill-posed, and we further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, one numerical example is given to show that the proposed numerical method is effective.