  In this paper, a non-polynomial spectral Petrov-Galerkin method and
associated collocation method for substantial fractional differential equations
(FDEs) are proposed, analyzed, and tested. We extend a class of generalized
Laguerre polynomials to form our basis. By a proper scaling of trial basis and
test basis, our Petrov-Galerkin method results in a diagonal and thus
well-conditioned linear systems for both fractional advection equation and
fractional diffusion equation. In the meantime, we construct substantial
fractional differential collocation matrices and provide explicit forms for
both type of equations. Moreover, the proposed method allows us to adjust a
parameter in basis selection according to different given data to maximize the
convergence rate. This fact has been proved in our error analysis and confirmed
in our numerical experiments.
