The Hermite Spectral Method for Gaussian-Type Functions

Although Hermite functions were widely used for many practical problems, numerical experiments with the standard (normalized) Hermite functions ψ ₙ (v) worked poorly in the sense that too many Hermite functions are required to solve differential equations. In order to obtain accurate numerical solutions, it is necessary to choose a scaling factor α and use ψ ₙ (α v) as the basis functions. In this paper the scaling factors are given for functions that are of Gaussian type, which have finite supports $[ - M,M]$. The scaling factor used is max 0 j N \ γ ⱼ \ / M, where \ γ ⱼ \ j = 0N are the roots of ψ N + 1 (v) and $N + 1$ is the number of the truncated terms used. The numerical results show that after using this scaling factor, only reasonable numbers of the Hermite functions are required to solve differential equations.