Spectral regularization method for the time fractional inverse advection–dispersion equation

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 problem is obtained from the classical advection–dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α(0<α<1). We show that the TFIADP is severely ill-posed and further apply a spectral 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, numerical examples are given to show that the proposed numerical method is effective.