A distributional approach to the boundary layer theory
type: Item
instance of: scholarly article
title: A distributional approach to the boundary layer theory
zbMATH Open document ID: 0836.34056
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-10-31T00:00:00Z
review text: The aim of this paper is to show that boundary layer theory for linear problems can be investigated using a distributional approach. As a first problem, the authors consider the solution \(y(x, \varepsilon)\) of the initial value problem \[ \varepsilon y^{(m)} + \sum^{m - 1}_{j = 0} a_j (x)y^{(j)} = f(x), \quad y^{(j)} (0) = y_j,\;j = 0, \ldots, m - 1, \] where \(\varepsilon\) is a small parameter. Under natural assumptions, they prove an expansion \[ z(x, \varepsilon) = H(x) y(x, \varepsilon) \sim \sum^\infty_{n = 0} z_n(x) \varepsilon^n , \] where \(H(x)\) is the Heavyside function and the \(z_n\) are in a distribution space. A second result refers to the possibility to split the expansion of \(z(x, \varepsilon) \sim z^{(ou)} (x, \varepsilon) + z^{(in)} (x, \varepsilon)\) into an outer and an inner expansion \[ z^{(ou)} (x, \varepsilon) \sim \sum^\infty_{n = 0} z_n^{(ou)} (x) \varepsilon^n, \quad z^{(in)} (x, \varepsilon) \sim \sum^\infty_{n = 0} z_n^{(in)} (x) \varepsilon^n, \] where the \(z_n^{(ou)}\) are regular and the \(z_n^{(in)}\) with support in \{0\}. A second problem concerns the boundary value problem \(\varepsilon y'' + a(x)y' + b(x)y = f(x)\), \(y(0) = A\), \(y(1) = B\). Introducing the auxiliary quantity \(C (\varepsilon) : = y' (0, \varepsilon)\), the authors obtain expansions \[ z(x, \varepsilon) = H(x) y(x, \varepsilon) \sim \sum^\infty_{n = 0} z_n (x) \varepsilon^n , \quad C (\varepsilon) \sim \sum^\infty_{n = - 1} C_n \varepsilon^n, \] where the distributions \(z_n\) are functions of \(C\). All these results are illustrated with various examples which clarify the scope of the abstract results.
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 811163
zbMATH Keywords: boundary layer
zbMATH Keywords: distributional approach
zbMATH Keywords: initial value problem
zbMATH Keywords: small parameter
zbMATH Keywords: expansion
zbMATH Keywords: Heavyside function
zbMATH Keywords: boundary value problem
author: R. Estrada
author: Ram P. Kanwal
reviewed by: Patrick Habets
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1204
OpenAlex ID: W1979079490
DOI: 10.1006/JMAA.1995.1204
description: scientific article; zbMATH DE number 811163