Analytical and Numerical Solutions of Riesz Space Fractional Advection-Dispersion Equations with Delay

In this paper, we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay (RFADED). We utilize the fractional backward differential formulas method of second order (FBDF2) and weighted shifted Grunwald difference (WSGD) operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED. Firstly, the FBDF2 and the shifted Grunwald methods are introduced. Secondly, based on the FBDF2 method and the WSGD operators, the finite difference method is applied to the problem. We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(kappa + h(2)) and O(kappa(2) + h(2)) respectively. Thirdly we find the analytical solution for RFDED in terms Mittag-Leffler type functions. Finally, some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.