Spectral regularization method for a Cauchy problem of the time fractional advection–dispersion equation

In this paper, a Cauchy problem for the time fractional advection–dispersion equation (TFADE) is investigated. 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 α ( 0 < α ≤ 1 ) . We show that the Cauchy problem of TFADE is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. The convergence estimate is obtained under a priori bound assumptions for the exact solution. Numerical examples are given to show the effectiveness of the proposed numerical method.