A numerical method for a time-fractional advection–dispersion equation with a nonlinear source term

Abstract

In this paper we propose an implicit finite-difference scheme to approximate the solution of an initial-boundary value problem for a time-fractional advection–dispersion equation with variable coefficients and a nonlinear source term. The time fractional derivative is taken in the sense of Caputo. The method is unconditionally stable and convergent. Some numerical examples are included and the results confirm the theoretical analysis. One of the examples is the fractional Fisher equation of mathematical biology.