  We present the Method Of Lines (MOL), which is based on the spectral
collocation method, to solve space-fractional advection-diffusion equations
(SFADEs) on a finite domain with variable coefficients. We focus on the cases
in which the SFADEs consist of both left- and right-sided fractional
derivatives. To do so, we begin by introducing a new set of basis functions
with some interesting features. The MOL, together with the spectral collocation
method based on the new basis functions, are successfully applied to the
SFADEs. Finally, four numerical examples, including benchmark problems and a
problem with discontinuous advection and diffusion coefficients, are provided
to illustrate the efficiency and exponentially accuracy of the proposed method.
