An RBF-FD method for the time-fractional advection–dispersion equation with nonlinear source term

Abstract

Fractional advection–dispersion equations have proved to be useful for modeling a wide range of problems in environmental and engineering sciences. In this work, we adapt a Radial Basis Function-generated Finite Difference (RBF-FD) method to obtain approximated numerical solutions of the initial-boundary value problem of the time-fractional advection–dispersion equation with variable coefficients and nonlinear source. We use a strategy of minimization of the local truncation error in approximating the initial condition to find appropriate local shape parameters for the Gaussian RBF. For discretizing the fractional time derivative, in the Caputo's sense, we use a scheme of (4−α)th-order, where α∈(0,1) is the order of the fractional derivative. We evaluate the performance of the RBF-FD method for different 2-dimensional problems on domains with complex geometries in which a high precision is exhibited of the solutions found. Particularly, we test our method for approximating solutions of Fisher’s equation with fractional time derivative, where the source that depends on the solution, is approximated by a linearization with respect to time, and we obtain a rate of convergence of second order in time.