On the Numerical Discretization of the Fractional Advection-Diffusion Equation with Generalized Caputo-Type Derivatives on Non-uniform Meshes

Abstract

A broad framework for time-fractional advection-diffusion equations is considered by incorporating a general class of Caputo-type fractional derivatives that includes many well-known fractional derivatives as particular cases. Within this general context, in this paper, we propose a numerical approach to provide approximate solutions to time-fractional advection-diffusion models that involve fractional derivatives of the generalized class. The developed approach is based on discretizing the studied models with respect to spatio-temporal domains using non-uniform meshes. Moreover, we discuss the stability of the proposed approach, which does not require solving large systems of linear equations. Numerical results and 3D graphics are presented for some time-fractional advection-diffusion models using several fractional derivatives. The feasibility and the reliability of the proposed approach are clearly demonstrated by comparing the numerical solutions of the studied models with the exact solutions in cases with a known solution.