On an internal boundary layer problem
type: Item
instance of: scholarly article
title: On an internal boundary layer problem
zbMATH Open document ID: 1012.34049
DOI: 10.1016/S0377-0427(01)00569-6
author: Heping Yang
author: Roderick Wong
published in: Journal of Computational and Applied Mathematics
publication date: 2002-08-22T00:00:00Z
review text: The authors consider the singular perturbation problem \[ \varepsilon y''+a(x)y'+b(x)y=0, \quad y(x_-)=A,\;y(x_+)=B, \tag{1} \] where \(x_-\leq 0\leq x_+\) and \(\varepsilon>0\) is a small parameter. They investigate a turning point problem such that \(a(x)=0\) if and only if \(x=0\), and near \(x=0\), \(a(x)\sim \alpha x\), \(b(x)\sim \beta\), with \(\alpha >0\) and \(\beta/\alpha\neq 1,2,\ldots\). The existence of a unique solution for small values of \(\varepsilon>0\) together with its asymptotic behavior is established. The idea of the approach is to use a change of variables so as to consider equation (1) as a perturbation of \[ \varepsilon u''+\alpha tu'+\beta u=0, \] the general solution to which can be written using parabolic cylinder functions. Problem (1) is then written as an integral equation and, for small values of \(\varepsilon\), its solution can be obtained using successive approximations.
Mathematics Subject Classification ID: 34E20
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 1787665
zbMATH Keywords: singular perturbation
zbMATH Keywords: turning points
zbMATH Keywords: internal boundary layer
reviewed by: Patrick Habets
MaRDI profile type: Publication
cites work: Q5558293
cites work: Boundary Layer Problems Exhibiting Resonance
cites work: Q3207312
cites work: Q4059363
cites work: Second-order linear differential equations with two turning points
cites work: Sufficient Conditions for Ackerberg–O’Malley Resonance
cites work: Singular perturbation methods for ordinary differential equations
cites work: On a Differential Equation of Boundary Layer Type
cites work: Q3682833
cites work: A singular perturbation problem with a turning point
description: scientific article; zbMATH DE number 1787665