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

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... ion equations and fractional nonlinear Schrödinger equations. ). A1928 SPECTRAL METHODS FOR FPDEs IN UNBOUNDED DOMAINS A1929 a different parameter, therefore, fractional derivatives become a local operator in the phase space spanned by fractopolynomials/GJFs. This property led to the very efficient spectral methods for fractional PDEs in bounded domains; see for instance [23, 24, 13, 6] . However, there is no apparent direct extension to fractional PDEs in unbounded domains. In this paper, we consider a special class of fractional PDEs in an unbounded domain which involves the fractional Laplacian operator (− ) α 2 defined through the Fourier transform [15] . While taking the fractional Laplacian under the Fourier transform is a simple and "local" operation, it is, however, very difficult to approximate the (continuous) Fourier transform. Attempts have been made in using the discrete Fourier transform on periodic domains (see, for instance, [11]), but it requires an exceedingly large number of unknowns to achieve a reasonable accuracy. A key observation for our approach is that the Hermite functions are eigenfunctions of the Fourier transform. This fact, together with the definition of the fractional Laplacian through the Fourier transform, makes the fractional Laplacian a local operator in the phase space expanded by Hermite functions. We shall first develop a Hermite-collocation method which is extremely simple to implement, followed by a Hermite-Galerkin method, which is more accurate than the Hermite-collocation method, but still very efficient. The rest of the paper is organized as follows. In the next section, we provide some preliminaries about Hermite functions and their approximation properties. In section 3, we present the Hermite-collocation method and derive corresponding error estimates. The Hermite-Galerkin method is considered in section 4. In section 5, we present numerical results for model elliptic equations and applications to fractional advection-diffusion equations and fractional nonlinear Schrödinger equations.