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 Uspekhi Mat. Nauk, 1976, Volume 31, Issue 6(192), Pages 102–122 (Mi umn4011)

The development of the theory of ordinary differential equations with a small parameter multiplying the highest derivative during the period 1966–1976

A. B. Vasil'eva

Abstract: We survey the results relating to the theory of singular perturbations, in particular, to the theory of ordinary differential equations with a small parameter multiplying the highest derivative, that have been obtained by pupils of A. N. Tikhonov during the period 1966-1976. The main attention is directed at problems in which the roots of the characteristic equation have real parts of different signs (the conditionally stable case), and also at problems in which the characteristic equation has roots that are identically zero.

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English version:
Russian Mathematical Surveys, 1976, 31:6, 109–131

Bibliographic databases:

UDC: 517.9
MSC: 34E15, 34E05, 15A18, 34Bxx, 34K10

Citation: A. B. Vasil'eva, “The development of the theory of ordinary differential equations with a small parameter multiplying the highest derivative during the period 1966–1976”, Uspekhi Mat. Nauk, 31:6(192) (1976), 102–122; Russian Math. Surveys, 31:6 (1976), 109–131

Citation in format AMSBIB
\Bibitem{Vas76} \by A.~B.~Vasil'eva \paper The development of the theory of ordinary differential equations with a~small parameter multiplying the highest derivative during the period 1966--1976 \jour Uspekhi Mat. Nauk \yr 1976 \vol 31 \issue 6(192) \pages 102--122 \mathnet{http://mi.mathnet.ru/umn4011} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=473373} \zmath{https://zbmath.org/?q=an:0366.34044|0376.34045} \transl \jour Russian Math. Surveys \yr 1976 \vol 31 \issue 6 \pages 109--131 \crossref{https://doi.org/10.1070/RM1976v031n06ABEH001580} 

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This publication is cited in the following articles:
1. R. E. O’Malley, Jr, “A Singular Singularly-Perturbed Linear Boundary Value Problem”, SIAM J Math Anal, 10:4 (1979), 695
2. J. J. Mahony, J. J. Shepherd, “Stiff systems of ordinary differential equations. Part 1. Completely stiff, homogeneous systems”, J Aust Math Soc Series B Appl Math, 23:1 (1981), 17
3. V. S. Korolyuk, “The Boundary Layer in the Asymptotic Analysis of Random Walks”, Theory Probab Appl, 34:1 (1989), 179
4. “Singular Perturbations and Time Scales in Guidance and Control of Aerospace Systems: A Survey”, Journal of Guidance, Control, and Dynamics, 24:6 (2001), 1057
5. Walter Kelley, “Perturbation problems with quadratic dependence on the first derivative”, Nonlinear Analysis: Theory, Methods & Applications, 51:3 (2002), 469
6. D. Subbaram Naidu, “Singular perturbations and hysteresis. Michael P. Mortell, Robert E. O'Malley, Alexei Pokrovskii and Vladimir Sobolev, Society of Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. No. of pages: xiv+344. Price: \$66.00. ISBN 0-089871-597-0”, Int J Robust Nonlinear Control, 17:12 (2007), 1155
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