Application of a Legendre collocation method to the space–time variable fractional-order advection–dispersion equation

Abstract

In this paper, an efficient spectral collocation method is presented for solving the space–time variable-order fractional advection–dispersion equations (ST-VO-FADE). The proposed method is based on the Legendre collocation spectral procedure together with the Legendre operational matrices for fractional derivatives, described in the sense of Riemann–Liouville and Caputo. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations, which greatly simplifies the solution process. The validity of the method is demonstrated by solving two numerical examples. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.

Keywords

• variable-order derivative
• space–time fractional advection–dispersion equation
• collocation method
• legendre polynomials