  Spectral discretizations of fractional derivative operators are examined,
where the approximation basis is related to the set of Jacobi polynomials. The
pseudo-spectral method is implemented by assuming that the grid, used to
represent the function to be differentiated, may not be coincident with the
collocation grid. The new option opens the way to the analysis of alternative
techniques and the search of optimal distributions of collocation nodes, based
on the operator to be approximated. Once the initial representation grid has
been chosen, indications on how to recover the collocation grid are provided,
with the aim of enlarging the dimension of the approximation space. As a
results of this process, performances are improved. Applications to fractional
type advection-diffusion equations, and comparisons in terms of accuracy and
efficiency are made. As shown in the analysis, special choices of the nodes can
also suggest tricks to speed up computations.
