Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem
type: Item
instance of: scholarly article
title: Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem
zbMATH Open document ID: 1160.34051
author: Zhiming Wang
author: Gexia Wang
author: W. Z. Lin
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-09-10T00:00:00Z
review text: The authors consider a two-point singularly perturbed boundary value problem on the interval \(t\in [0,1]\) for a system of differential equations \[ \varepsilon\frac{dz}{dt}=f(z,y,t),\qquad \frac{dy}{dt}=g(z,y,t), \] where \(z\) and \(f\) are both \(M\)-dimensional vectors, \(y\) and \(g\) are both \(m\)-dimensional vectors. Boundary conditions for unknown vector-function \(y\) are given at the left interval's end only (\(y(0,\varepsilon)=y^0\)), while boundary conditions for unknown vector-function \(z\) are really two-point boundary condition (\(a z(0,\varepsilon)+b z(1,\varepsilon)=z^0\), with \(a\) and \(b\) being constant matrices). It is well-known that under certain general assumptions referred as the case of conditional stability, the above-formulated problem has a solution with an asymptotics given by boundary function method. The main result of this paper is that the latter solution is continuously differentiable with respect to the boundary data \(y^0\) and \(z^0\). Moreover, every derivative with respect to the boundary data is well-approximated by the corresponding derivative of the asymptotics obtained via boundary function method.
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 5324195
zbMATH Keywords: differentiability
zbMATH Keywords: singular perturbation
zbMATH Keywords: boundary function method
zbMATH Keywords: conditional stability
reviewed by: Oleh Omel'chenko
MaRDI profile type: Publication
full work available at URL: j.na.2007.08.004
OpenAlex ID: W1980256857
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DOI: 10.1016/J.NA.2007.08.004
description: scientific article; zbMATH DE number 5324195