The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation

In this paper, we consider a Riesz fractional advection–dispersion equation (RFADE), which is derived from the kinetics of chaotic dynamics. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives by the Riesz fractional derivatives of order α ∈ (0, 1) and β ∈ (1, 2], respectively. We derive the fundamental solution for the Riesz fractional advection–dispersion equation with an initial condition (RFADE-IC). We investigate a discrete random walk model based on an explicit finite-difference approximation for the RFADE-IC and prove that the random walk model belongs to the domain of attraction of the corresponding stable distribution. We also present explicit and implicit difference approximations for the Riesz fractional advection–dispersion equation with initial and boundary conditions (RFADE-IBC) in a finite domain. Stability and convergence of these numerical methods for the RFADE-IBC are discussed. Some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.