Conclusions and Further Developments

Abstract

Spectral tau, Galerkin and collocation methods are briefly revised. On bounded domains all of them are constructed using Chebyshev polynomials. Collocation on an unbounded domain is based on Laguerre functions. Practical and computational aspects of these methods are mainly emphasized. High order eigenvalue problems, i.e., of fourth, sixth and eighth orders along with genuinely nonlinear and singular perturbed two-point eigenvalue problems are considered. The capabilities of the methods are analysed based on the conditioning and normality of the differentiation matrices in both the physical and phase (coefficient) spaces.

Applied mathematics is subject to fads and enthusiasms. We have shown above that the theory of numerical algorithms contains hidden beyond-all-orders terms, but this aspect of numerical analysis is largely terra incognita.

John P. Boyd, The Devil’s Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series, 2000.