Padé–Gegenbauer suppression of Runge phenomenon in the diagonal limit of Gegenbauer approximations

In this paper we present a random walk model for approximating a Lévy–Feller advection–dispersion process, governed by the Lévy–Feller advection–dispersion differential equation (LFADE). We show that the random walk model converges to LFADE by use of a properly scaled transition to vanishing space and time steps. We propose an explicit finite difference approximation (EFDA) for LFADE, resulting from the Grünwald–Letnikov discretization of fractional derivatives. As a result of the interpretation of the random walk model, the stability and convergence of EFDA for LFADE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.