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

Abstract

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 $p\in [0,1]$; 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, wh...