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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2019, Volume 29, Issue 3, Pages 332–340 (Mi vuu686)  

MATHEMATICS

Asymptotics of the solution to the boundary-value problem when the limit equation has an irregular singular point

K. G. Kozhobekova, D. A. Tursunovba

a Osh State University, ul. Lenina, 331, Osh, 723500, Kyrgyzstan
b Osh Branch of the Russian State Social University, ul. Karasuiskaya, 161, Osh, 723506, Kyrgyzstan

Abstract: This article studies the asymptotic behavior of the solutions of singularly perturbed two-point boundary value-problems on an interval. The object of the study is a linear inhomogeneous ordinary differential second-order equation with a small parameter with the highest derivative of the unknown function. The special feature of the problem is that the small parameter is found at the highest derivative of the unknown function and the corresponding unperturbed first-order differential equation has an irregular singular point at the left end of the segment. At the ends of the segment, boundary conditions are imposed. Two problems are considered: in one of them the function in front of the first derivative of the unknown function is nonpositive on the segment considered, and in the second it is nonnegative. Asymptotic expansions of the problems are constructed by the classical method of Vishik–Lyusternik–Vasilyeva–Imanaliev boundary functions. However, this method cannot be applied directly, since the external solution has a singularity. We first remove this singularity from the external solution, and then apply the method of boundary functions. The constructed asymptotic expansions are substantiated using the maximum principle, i.e., estimates for the residual functions are obtained.

Keywords: irregular singular point, singular perturbation, asymptotic behavior, methods of boundary layer functions, Dirichlet problem, boundary function, small parameter.

Funding Agency Grant Number
Ministry of Education and Science of Kyrgyz Republic
The research was funded partially by Ministry of Education and Science of the Kyrgyz Republic.


DOI: https://doi.org/10.20537/vm190304

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Bibliographic databases:

UDC: 517.928.2
MSC: 34E05, 34E10, 34E20, 34B05
Received: 11.05.2019

Citation: K. G. Kozhobekov, D. A. Tursunov, “Asymptotics of the solution to the boundary-value problem when the limit equation has an irregular singular point”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:3 (2019), 332–340

Citation in format AMSBIB
\Bibitem{KozTur19}
\by K.~G.~Kozhobekov, D.~A.~Tursunov
\paper Asymptotics of the solution to the boundary-value problem when the limit equation has an irregular singular point
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2019
\vol 29
\issue 3
\pages 332--340
\mathnet{http://mi.mathnet.ru/vuu686}
\crossref{https://doi.org/10.20537/vm190304}


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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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