# Singularly Perturbed Ordinary Differential Equation with Turning Point and Interior Layer

- Authors
- 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
- License
- Yellow

## 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.