Polynomial Spectral collocation Method for Space Fractional Advection-Diffusion Equation

This paper discusses the spectral collocation method for numerically solving nonlocal problems: one dimensional space fractional advection-diffusion equation; and two dimensional linear/nonlinear space fractional advection-diffusion equation. The differentiation matrixes of the left and right Riemann-Liouville and Caputo fractional derivatives are derived for any collocation points within any given interval. The stabilities of the one dimensional semi-discrete and full-discrete schemes are theoretically established. Several numerical examples with different boundary conditions are computed to testify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy-Feller advection-diffusion equation are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases.