  This article studies a direct numerical approach for fractional
advection-diffusion equations (ADEs). Using a set of cubic trigonometric
B-splines as test functions, a differential quadrature (DQ) method is firstly
proposed for the 1D and 2D time-fractional ADEs of order $(0,1]$. The weighted
coefficients are determined, and with them, the original equation is
transformed into a group of general ordinary differential equations (ODEs),
which are discretized by an effective difference scheme or Runge-Kutta method.
The stability is investigated under a mild theoretical condition. Secondly,
based on a set of cubic B-splines, we develop a new Crank-Nicolson type DQ
method for the 2D space-fractional ADEs without advection. The DQ
approximations to fractional derivatives are introduced and the values of the
fractional derivatives of B-splines are computed by deriving explicit formulas.
The presented DQ methods are evaluated on five benchmark problems and the
concrete simulations of the unsteady propagation of solitons and Gaussian
pulse. In comparison with the existing algorithms in the open literature,
numerical results finally illustrate the validity and accuracy.
