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. © 2009 Elsevier B.V. All rights reserved.
Zheng, G. H., & Wei, T. (2010). Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation. Journal of Computational and Applied Mathematics, 233(10), 2631–2640. https://doi.org/10.1016/j.cam.2009.11.009