Vestnik Yuzhno-Ural'skogo Gosudarstvennogo Universiteta. Seriya "Matematika. Mekhanika. Fizika"
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Vestnik Yuzhno-Ural'skogo Gosudarstvennogo Universiteta. Seriya "Matematika. Mekhanika. Fizika", 2016, Volume 8, Issue 2, Pages 52–61
DOI: https://doi.org/10.14529/mmph160207
(Mi vyurm299)
 

This article is cited in 4 scientific papers (total in 4 papers)

Asymptotics of the solution to the bisingular perturbed Dirichlet problem in the ring with quadratic growth on the boundary

D. A. Tursunova, U. Z. Erkebaevb

a Urals State Pedagogical University, Ekaterinburg, Russian Federation
b Osh State University, Osh, Kyrgyzstan
Full-text PDF (438 kB) Citations (4)
References:
Abstract: The Dirichlet problem for elliptic equations with a small parameter in the highest derivatives takes a unique place in mathematics. In general case it is impossible to build explicit solution to these problems, which is why the researchers apply different asymptotic methods. The aim of the research is to develop the asymptotic method of boundary functions for constructing complete asymptotic expansions of the solutions to such problems. The proposed generalized method of boundary functions differs from the matching method in the fact that the growing features of the outer expansion are actually removed from it and with the help of the auxiliary asymptotic series are fully included in the internal expansions, and differs from the classical method of boundary functions in the fact that the boundary functions decay in power-mode nature and not exponentially. Using the proposed method, a complete asymptotic expansion of the solution to the Dirichlet problem for bisingular perturbed linear inhomogeneous second-order elliptic equations with two independent variables in the ring with quadratic growth on the boundary is built. A built asymptotic series corresponds to the Puiseux series. The basic term of the asymptotic expansion of the solution has a negative fractional degree of the small parameter, which is typical for bisingular perturbed equations, or equations with turning points. The built expansion is justified by the maximum principle.
Keywords: asymptotic expansion of a solution, bisingular perturbation, elliptic equation, Dirichlet problem in the ring, small parameter, generalized method of boundary functions, boundary functions, modified Bessel functions.
Received: 24.02.2016
Bibliographic databases:
Document Type: Article
UDC: 517.955.8
Language: Russian
Citation: D. A. Tursunov, U. Z. Erkebaev, “Asymptotics of the solution to the bisingular perturbed Dirichlet problem in the ring with quadratic growth on the boundary”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 8:2 (2016), 52–61
Citation in format AMSBIB
\Bibitem{TurErk16}
\by D.~A.~Tursunov, U.~Z.~Erkebaev
\paper Asymptotics of the solution to the bisingular perturbed Dirichlet problem in the ring with quadratic growth on the boundary
\jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz.
\yr 2016
\vol 8
\issue 2
\pages 52--61
\mathnet{http://mi.mathnet.ru/vyurm299}
\crossref{https://doi.org/10.14529/mmph160207}
\elib{https://elibrary.ru/item.asp?id=25903366}
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  • https://www.mathnet.ru/eng/vyurm/v8/i2/p52
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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    Full-text PDF :48
    References:49
     
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