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Two-Timing, Geometric, and Multi-scale Methods

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Historical Developments in Singular Perturbations

Abstract

The Brooklyn native Julian Cole (1925–1999) got his Ph.D. in aeronautics at Caltech in 1949 with Hans Liepmann (a German émigre of 1939) as his advisor. He remained on the Caltech faculty until 1968 where he maintained active contact with Kaplun, Lagerstrom and others in aeronautics, applied mathematics and industry, attempting to understand singular perturbations more deeply and to apply its growing methodology. Then he moved to UCLA and, ultimately, Rensselaer. He and his Jerusalem-born student Jerry Kevorkian , who spent his academic career at the University of Washington, developed and applied asymptotic methods involving two- (i.e. multi-) time or multiple scales in the early 1960s (cf. the obituary of Cole by Bluman et al. [48]). Related approaches were made by the Soviet Kuzmak [273], the Australian Mahony [305], and the American Cochran [89], among others, but Cole and Kevorkian had the dominant long-term impact. The previously cited work of Lomov is also recommended reading, as is the paper by Levey and Mahony [286]. The monograph Perturbation Methods in Applied Mathematics , Cole [92], considered singular perturbations in a broad applied math setting, where both the development of the underlying techniques and significant and diverse applications were included. The book approaches matching using intermediate limits and presumes a corresponding overlap of inner and outer domains. The examples used are generally very instructive and quite nontrivial.

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O’Malley, R.E. (2014). Two-Timing, Geometric, and Multi-scale Methods. In: Historical Developments in Singular Perturbations. Springer, Cham. https://doi.org/10.1007/978-3-319-11924-3_5

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