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The Hermite spectral method for Gaussian-type functions. (English) Zbl 0782.65110
Although Hermite functions are widely used for approximating the solutions of ordinary differential equations, numerical experiments with the standard (normalized) Hermite functions indicate that these functions can be inefficient. Efficiency can be improved by choosing a scaling parameter \(\alpha\) and using \(\psi_ n(\alpha x)\) as the basis functions.
In this paper the scaling factors are given for functions of Gaussian type, having finite support in \([-M,M]\). The scaling factor used is \(\max_ j\gamma_ j/M\), where the \(\gamma_ j\) are the zeros of \(\psi_{N+1}\), and \(N+1\) is the number of truncated terms used.

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
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