A Spectral Method (of Exponential Convergence) for Singular Solutions of the Diffusion Equation with General Two-Sided Fractional Derivative

An open problem in the numerical solution of fractional partial differential equations (FPDEs) is how to obtain high-order accuracy for singular solutions; even for smooth right-hand sides solutions of FPDEs are singular. Here, we consider the one-dimensional diffusion equation with general two-sided fractional derivative characterized by a parameter p∈ [0,1]; for $p=1/2$ we recover the Riesz fractional derivative, while for $p = 1$, 0 we obtain the one-sided fractional derivative. We employ a Petrov--Galerkin projection in a properly weighted Sobolev space with (two-sided) Jacobi polyfracnomials as basis and test functions. In particular, we derive these two-sided Jacobi polyfractonomials as eigenfunctions of a Sturm--Liouville problem with weights uniquely determined by the parameter p. We provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term, i.e., for smooth data (corresponding to singular solutions) we obtain exponential convergence, while for smooth solutions we obtain algebraic convergence. We demonstrate the sharpness of our error estimates with numerical examples, and we present comparisons with a competitive spectral collocation method of tunable accuracy. We also investigate numerically deviations from the theory for inhomogeneous Dirichlet boundary conditions as well as for a fractional diffusion-reaction equation.