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Basic Concepts Underlying Singular Perturbation Techniques

Lagerstrom, P. A. and Casten, R. G. (1972) Basic Concepts Underlying Singular Perturbation Techniques. SIAM Review, 14 (1). pp. 63-120. ISSN 0036-1445. doi:10.1137/1014002.

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In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate x^* = ε^(-1)x. In a secular-type problem x and x^* are used simultaneously. This paper discusses layer-type problems in which x^* is used in a thin layer and x outside this layer. Assume one seeks approximations to a function f(x,ε), uniformly valid to some order in ε for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define "domain of validity" one needs to consider intervals whose endpoints depend on epsilon. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions "matching." The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions.

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Additional Information:©1972 Society for Industrial and Applied Mathematics. Received by the editors November 24, 1970, and in final revised form April 19, 1971. This invited paper was prepared in part and published under Contract DA-49-092-ARO-110 with the U.S. Army Research Office. The research was supported in part by the National Science Foundation under Grant GP28129.
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Deposited On:24 Jun 2008
Last Modified:08 Nov 2021 21:13

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