Hermite spectral collocation methods for fractional PDEs in unbounded domains

TL;DR

This paper develops Hermite spectral collocation methods using Hermite-type basis functions for solving fractional PDEs in unbounded domains, with recursive differentiation matrices and applications to multi-term equations.

Contribution

It introduces new spectral collocation techniques with Hermite basis functions and recursive differentiation matrices for fractional PDEs in unbounded domains.

Findings

Effective spectral methods demonstrated through numerical examples

Addresses condition number and scaling issues in spectral collocation

Applicable to multi-term fractional PDEs

Abstract

This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss-Hermite points. In this work, two Hermite-type functions are employed to serve as basis functions. Our main task is to find corresponding differentiation matrices which are computed recursively. Two important issues relevant to condition numbers and scaling factors will be discussed. Applications of the spectral collocation methods to multi-term fractional PDEs are also presented. Several numerical examples are carried out to demonstrate the effectiveness of the proposed methods.