Rational Spectral Methods for PDEs Involving Fractional Laplacian in Unbounded Domains

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decaying slowly and subject to certain power law. Their numerical solutions are underexplored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identities related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by precomputing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach.