Singularly perturbed problems with double singularity
type: Item
instance of: scholarly article
title: Singularly perturbed problems with double singularity
zbMATH Open document ID: 0921.34056
DOI: 10.1007/BF02358973
author: Yu. A. Konyaev
published in: Mathematical Notes
publication date: 1997-01-01T00:00:00Z
review text: The author deals with systems of linear differential equations \[ \varepsilon^{m_1}\bigl(\delta_0(x)+\varepsilon\bigr)^{m_2}{dy\over dx}=A(x,\varepsilon)y+f(x), \] with \(\delta_0(x)>0\) for \(x\in(0,1)\), \(\delta_0 (0)=\delta_0(1)=0\), \(m_1,m_2\) are positive integers (which determine the two singularities \(x=0\) and \(x=1\), respectively) and the matrix \(A(x,\varepsilon)\) possesses an absolutely and uniformly convergent expansion \(\sum^\infty_{k=0} A_k(x) \varepsilon^k\), in which \(A_0(x)\) has a simple nonzero stable spectrum. Asymptotic expansions for a bounded (as \(\varepsilon\to 0)\) solution \(y(x, \varepsilon)\) are proved in two theorems. In contrast to usual results, the structure of these asymptotic formulas does not depend only on the spectrum of \(A_0(x)\), even if \(m_1=m_2=1\).
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 1257134
zbMATH Keywords: singular perturbations
zbMATH Keywords: boundary value problems
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: linear differential equations
reviewed by: J. Šimša
MaRDI profile type: Publication
cites work: Q5522440
cites work: Q3362809
cites work: ON A METHOD OF STUDYING CERTAIN PROBLEMS IN PERTURBATION THEORY
full work available at URL: https://doi.org/10.1007/bf02358973
OpenAlex ID: W2069303244
description: scientific article; zbMATH DE number 1257134