On Singular Perturbations, Especially Matching

Abstract

Among fifty very strong minisymposia at ICIAM 91 (cf. O'Malley (1992)), that on the Historical Development of Singular Perturbation Concepts, somewhat surprisingly, seemed to be the most popular. In retrospect, this was certainly due to both its accomplished participants and to the importance of asymptotic analysis throughout contemporary applied mathematics. As a field like singular perturbations matures, newcomers often learn much about recent progress and skillfully use accumulated methods without giving any thought to those people and problems that were fundamental to its development. As national, distance, and disciplinary barriers become lowered, it is perhaps less easy to comprehend how isolated earlier investigators were, yet how critical was scientific contact and interchange with others working elsewhere. Although considerable current research uses an expanding menu of singular perturbation methods in varied computational and initerdisciplinary contexts, many fundamental aspects of the theory were developed worldwide in the twenty years after the second world war. They now comprise basic elements in the curriculum for many young applied mathematicians, engineers, and scientists. Those who anticipated and initiated this substantial effort are now no longer active. It would certainly be provocative and revealing to listen to a round-table discussion on contemporary asymptotics involving major figures like Birkhoff, Friedrichs, Goldstein, Lagerstrom, Langer, Levinson, Mitropolskii, Prandtl, Tikhonov, van der Pol, von Karman, and Wasow. The participants in our 1991 symposium have all written expanded versions of their talks which pleasantly display both brilliant and idiosyncratic insights of the author. A critic could easily point out a number of perspectives not represented. Thus, I am sure that the authors join me in hoping that this collection of essays will prove useful to those who further develop and apply the basic ideas of singular perturbations to new and significant scientific problems and that it will encourage others to learn and write about the history of this fascinating intellectual undertaking. The literature does, indeed, include historical discussions (cf., e.g., Bohnenblust et al. (1953), Wilcox (1964), Kaplun (1967), Vasil'eva and Volosov (1967), Goldstein (1969), McHugh (1971), Tani (1977), Sanders and Verhulst (1985), Wasow (1985), Morawetz (1986), Lagerstrom (1988), O'Malley (1991), and Yamaguti et al. (1993)). These papers emphasize the asymptotic matching of approximations: rudimentary nineteenth-century examples arising in many fields, expansion procedures for the NavierStokes equations for high and low Reynolds numbers, debates about whether matching needs an overlap hypothesis, and the simpler boundary function (or boundary layer correction) method which can often be simply employed. That matching remains central to singular perturbations theory and that it continues to provide challenges in practice is obvious from the recent monograph, by Il'in (1992). The authors of the four papers which follow join me in hoping that you enjoy them and learn from them.