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 psi(n)(upsilon) 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 alpha and use psi(n)(alphaupsilon) 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 max0 less-than-or-equal-to j less-than-or-equal-to N {gamma(j)}/M, where {gamma(j)}j=0N are the roots of psi(N+1)(upsilon) 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.