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Kyrgyzstan; Russia Osh; Yekaterinburg
Section  Mathematics 
Title  Asymptotics of the Dirichlet problem solution for a bisingular perturbed equation in the ring 
Author(s)  Tursunov D.A.^{ab}, Erkebaev U.Z.^{b} 
Affiliations  Ural State Pedagogical University^{a}, Osh State University^{b} 
Abstract  The paper refers to the asymptotic behavior of the Dirichlet problem solution for a bisingular perturbed elliptic secondorder equation with two independent variables in the ring. To construct the asymptotic expansion of the solution the authors apply the modified scheme of the method of boundary functions by VishikLyusternikVasil'evaImanaliev. The proposed method differs from the matching method by the fact that growing features of the outer expansion are in fact removed from it and with the help of an auxiliary asymptotic series are placed entirely in the internal expansion, and from the classical method of boundary functions by the fact that boundary functions have powerlaw decrease, not exponential. An asymptotic expansion of the solution is a series of Puiseux. The resulting asymptotic expansion of the Dirichlet problem solution is justified by the maximum principle. 
Keywords  formal asymptotic expansion, Dirichlet problem, Airy function, Puiseux series, small parameter, method of boundary functions, bisingular perturbation 
UDC  517.955.8 
MSC  35J25, 35J75, 35J15 
DOI  10.20537/vm150408 
Received  13 October 2015 
Language  Russian 
Citation  Tursunov D.A., Erkebaev U.Z. Asymptotics of the Dirichlet problem solution for a bisingular perturbed equation in the ring, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 4, pp. 517525. 
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