Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T20:32:15.885Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Whittaker, E. T.Proc. Roy. Soc. Edinb. 35 (1915), 181.Google Scholar
(2)Rutherford, D. E.Proc. Roy. Soc. Edinb. 62 (1947), 229;Google Scholar
Rutherford, D. E.Proc. Roy. Soc. Edinb. 63 (1952), 232.Google Scholar
(3)Todd, John.Proc. Camb. Phil. Soc. 46 (1950), 116.CrossRefGoogle Scholar
(4)Bromwich, T. J. I'A.An introduction to the theory of infinite series (London, 1926).Google Scholar
(5)Lanczos, C.Applied analysis (London, 1957).Google Scholar
(6)de la Vallée Poussin, Ch.-J.Acad. Roy. Belg. Bull. Cl. Sci. (1908), 319.Google Scholar
(7)Dennis, S. C. R. and Poots, G.Proc. Camb. Phil. Soc. 51 (1955), 422.CrossRefGoogle Scholar