Approximate solutions for the fractional advection-dispersion equation using Legendre pseudo-spectral method

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. Fractional advection-dispersion equation (FADE) is used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium and for modeling transport at the Earth surface. In this paper, an efficient numerical method for solving FADE is considered. The fractional derivative is described in the Caputo sense. The method is based on Legendre approximations. The properties of Legendre polynomials are utilized to reduce FADE to a system of ODEs, which is solved using the finite difference method. Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of FADE are presented and the results are compared with the exact solution.