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On asymptotic approximations of solutions of an equation with a small parameter


Authors: A. M. Il′in and E. F. Lelikova
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 927-939
MSC (2010): Primary 35J47
DOI: https://doi.org/10.1090/S1061-0022-2011-01177-X
Published electronically: August 19, 2011
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Abstract: A second order elliptic equation with a small parameter at one of the highest order derivatives is considered in a three-dimensional domain. The limiting equation is a collection of two-dimensional elliptic equations in two-dimensional domains depending on one parameter. By the method of matching of asymptotic expansions, a uniform asymptotic approximation of the solution of a boundary-value problem is constructed and justified up to an arbitrary power of a small parameter.


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Additional Information

A. M. Il′in
Affiliation: Chelyabinsk State University, Ul. Brat′yev Kashirinyh 129, Chelyabinsk 454001, Russia
Email: iam@csu.ru

E. F. Lelikova
Affiliation: Institute of Mathematics and Mechanics, of Ural Branch of Russian Academy of Science, Ul. Sof′i Kovalevskoi 16, Ekaterinburg 620990, Russia
Email: lef@imm.uran.ru

DOI: https://doi.org/10.1090/S1061-0022-2011-01177-X
Keywords: Asymptotic, boundary value problem, small parameter, matching of asymptotic expansions
Received by editor(s): June 11, 2010
Published electronically: August 19, 2011
Additional Notes: Supported by RFBR (grants nos. 08-01-00260, 09-01-00530), by a grant NSh-6249.2010.1, and by a grant FTsP 02.740.110612
Dedicated: Dedicated to our teacher Vasiliĭ Mikhaĭlovich Babich
Article copyright: © Copyright 2011 American Mathematical Society