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Singularly Perturbed Ordinary Differential Equation with Turning Point and Interior Layer

Authors
  • Tursunov, D. A.1
  • Sulaimanov, Z. M.1
  • Khalmatov, A. A.2
  • 1 Osh State University, Kyrgyz-Russian Faculty, Osh, 723500, Kyrgyzstan , Osh (Kyrgyzstan)
  • 2 Kyrgyz-Uzbek International University, Department of Mathematics, Osh, 723500, Kyrgyzstan , Osh (Kyrgyzstan)
Type
Published Article
Journal
Lobachevskii Journal of Mathematics
Publisher
Pleiades Publishing
Publication Date
Dec 13, 2021
Volume
42
Issue
12
Pages
3016–3021
Identifiers
DOI: 10.1134/S1995080221120362
Source
Springer Nature
Keywords
Disciplines
  • Article
License
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Abstract

AbstractThe article investigates the two point boundary-value problem for a singularly perturbed linear in homogeneous ordinary differential equation of the second order on a segment with a turning point and interior layer. The aim of the study is to construct a uniform asymptotic expansion of the solution of the two point problem with an arbitrary degree of accuracy on the considered segment, when a small parameter tends to zero. Features of the task: the small parameter is present before the highest derivative of the unknown functions; the coefficient function before the first-order derivative of the unknown function changes its sign from positive to negative; singularly perturbed equation has a turning point at the inner point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0$$\end{document}, an inner layer appears in the vicinity of this point. The complete asymptotic solution of the two point problem is constructed by the modified method of the boundary functions of Goldenveizer–Vishik–Lyusternik–Vasilyeva–Imanalieva. The asymptotic solution by the maximum principle is justified.

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