Search

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving an integral fractional Laplacian in Rd, which is built upon two essential components: (i) the Dunford-- Taylor formulation of the fractional Laplacian; and (ii) Fourier-like biorthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalized, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., O((N log2 N)d) with N being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schrödinger equation by using the fourth-order time-splitting method together with the proposed MCF--spectral-Galerkin method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]