Spectral method for substantial fractional differential equations

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.