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Eigenvalues and eigenfunctions of finite-difference operators

Published online by Cambridge University Press:  24 October 2008

W. G. Bickley  and
John McNamee
W. G. Bickley
Affiliation:
Imperial CollegeLondon
John McNamee
Affiliation:
University of AlbertaEdmonton, Canada
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Extract

Numerical solution of differential and integral equations is concerned mainly with the determination of the wanted function at a finite number of discrete points which are, in general, uniformly spaced. A first approximation to the solution can be obtained if the given differential or integral system is replaced by a difference system. Any differential or integral operator can be expressed as an infinite series of difference operators and the difference system of the first approximation is obtained by neglecting all but the first few terms of the infinite expansions. We shall distinguish two processes for improving the approximation: the first uses a tabular interval of moderate length but the approximation to the given continuous system is improved by including in the difference system as many terms of the infinite expansions as are necessary or practicable; the second process uses an unvarying difference system of low accuracy, but the tabular interval is reduced in successive approximations, the process being continued until two successive approximations agree within the accuracy required. We regard these processes as essentially distinct. If the solutions obtained by the two processes approach limits, these limits need not coincide.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 3 , July 1961 , pp. 532 - 546
Copyright
Copyright © Cambridge Philosophical Society 1961

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