Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-29T21:06:24.695Z Has data issue: false hasContentIssue false
  • English
  • Français

On the method of matched asymptotic expansions

Part I: A matching principle

Published online by Cambridge University Press:  24 October 2008

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The method of matched (or of ‘inner and outer’) asymptotic expansions is reviewed, with particular reference to two general techniques which have been proposed for ‘matching’; that is, for establishing a relationship between the inner and outer expansions, to finite numbers of terms, of an unknown function. It is shown that the first technique, which uses the idea of overlapping of the two expansions, can be difficult and laborious in some applications; while the second, which is the ‘asymptotic matching principle’ in the form stated by Van Dyke(13) can be incorrect. Two different sets of conditions sufficient for the validity of the asymptotic matching principle are then established, on the basis of assumptions about the structure of expressions which approximate to the desired function f(x,∈) for all relevant values of x. Finally, it is noted that in four classes of singular-perturbation problems for which complete and rigorous asymptotic theories exist, uniform approximations to the solutions have a structure which is a particular form of the general one assumed in this paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 1 , January 1969 , pp. 209 - 231
Copyright
Copyright © Cambridge Philosophical Society 1969

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)Eckhaus, W. and de Jager, . Arch. Rational Mech. Anal. 23 (1966), 26–86.CrossRefGoogle Scholar
(2)Erdélyi, A.Atti Accad. Sci. Torino 95 (1961), 651–672.Google Scholar
(3)Friedrichs, K. O.Special topics in analysis. Lecture notes, New York University (1954).Google Scholar
(4)Friedrichs, K. O.Bull. Amer. Math. Soc. 61 (1955), 485–504.CrossRefGoogle Scholar
(5)Hardy, G. H.Orders of infinity (Cambridge University Press, 1924).Google Scholar
(6)Kaplun, S.Z. Angew. Math. Physik 5 (1954), 111–135.CrossRefGoogle Scholar
(7)Kaplun, S.J. Math. Mech. 6 (1957), 595–603.Google Scholar
(8)Kaplun, S. and Lagerstrom, P. A.J. Math. Mech. 6 (1957), 585–593.Google Scholar
(9)Lagerstrom, P. A.J. Math. Mech. 6 (1957), 605–606.Google Scholar
(10)Lagerstrom, P. A. and Cole, J. D.J. Rational Mech. Anal. 4 (1955), 817–882.Google Scholar
(11)Lagerstrom, P. A., Howard, L. N. and Liu, C.-S. (Editors). Fluid mechanics and singular perturbations, a collection of papers by Saul Kaplun (New York: Academic Press, 1967).Google Scholar
(12)Proudman, I. and Pearson, J. R. A.J. Fluid Mech. 2 (1957), 237–262.CrossRefGoogle Scholar
(13)Van Dyke, M. D.Perturbation methods in fluid mechanics (New York: Academic Press, 1964).Google Scholar
(14)Vasil'eva, A. B.Uspehi Mat. Nauk 18 (1963), no. 3, 15–86. Translated in Russian Math. Surveys 18, no. 3, 13.Google Scholar
(15)Višik, M.I. and Lyusternik, L. A.Uspehi Mat. Nauk. 12 (1957), no. 5 (77), 3–122. Translated in Amer. Math. Soc. Transl., series 2, 20, 239.Google Scholar
(16)Wasow, W.J. Math. and Phys. Mass. Inst. Tech. 23 (1944), 173–183.Google Scholar
(17)Wasow, W.Asymptotic expansions for ordinary differential equations (New York: Interscience, Wiley, 1965).Google Scholar