Hermite Spectral Methods for Fractional PDEs in Unbounded Domains

Numerical approximations of fractional PDEs in unbounded domains are considered in this paper. Since their solutions decay slowly with power laws at infinity, a domain truncation approach is not effective as no transparent boundary condition is available. We develop efficient Hermite-collocation and Hermite--Galerkin methods for solving a class of fractional PDEs in unbounded domains directly, and derive corresponding error estimates. We apply these methods for solving fractional advection-diffusion equations and fractional nonlinear Schrödinger equations.