Hermite Spectral Collocation Methods for Fractional PDEs in Unbounded Domains

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, a...

Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains

Hermite spectral methods are investigated for linear diffusion equations and the viscous Burgers’ equation in unbounded domains. When the solution domain is unbounded, the diffusion operator no longer has a compact resolvent, which makes the Hermite spectral methods unstable. To overcome this difficulty, a time-dependent scaling factor is employed in the Hermite expansions, which yields a posit...

SOLVING SINGULAR ODES IN UNBOUNDED DOMAINS WITH SINC-COLLOCATION METHOD

Spectral approximations for ODEs in unbounded domains have only received limited attention. In many applicable problems, singular initial value problems arise. In solving these problems, most of numerical methods have difficulties and often could not pass the singular point successfully. In this paper, we apply the sinc-collocation method for solving singular initial value problems. The ability...

A Short Course on: Spectral Methods for Fractional ODEs/PDEs

Fractional Spectral Collocation Method

We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri ...