  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 L\'evy-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.
