Asymptotic properties of formal fundamental matrices of systems of second-order linear differential equations that contain a parameter
type: Item
instance of: scholarly article
title: Asymptotic properties of formal fundamental matrices of systems of second-order linear differential equations that contain a parameter
zbMATH Open document ID: 0531.34042
DOI: 10.1007/BF01093179
author: N. I. Shkil'
author: I. M. Konet
published in: Ukrainian Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Consider the second order linear system \(\epsilon^{p/q}\ddot x+A(t,\epsilon)x=0\) where \(\epsilon>0\) is a small parameter, p, q are arbitrary natural numbers and A(t,\(\epsilon)\) is a real \(n\times n\) matrix, represented by a convergent series \(A(t,\epsilon)=\sum^{\infty}_{0}\epsilon^ rA_ r(t).\) A formal fundamental matrix of the system is a matrix X(t,\(\epsilon)\) which verifies identically the differential equation in the sense of formal matrix series equality. It is shown that if the matrices \(A_ r(t)\) are \(C^{\infty}\) on [0,L] and the eigenvalues of \(A_ 0(t)\) are of constant multiplicity for 0\(\leq t\leq L\), their elementary divisors being also of constant multiplicity for 0\(\leq t\leq L\) then there exists a formal fundamental matrix \(X(t,\epsilon)=U(t,\epsilon)H(t,\epsilon)\) where \(0<\epsilon<\epsilon_ 0\), 0\(\leq t\leq L\) and U(t,\(\epsilon)\), H(t,\(\epsilon)\) can be computed as formal series with respect to the fractionary powers of \(\epsilon\).
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34A25
zbMATH DE Number: 3842311
zbMATH Keywords: second-order linear differential equations
zbMATH Keywords: slowly varying coefficients
zbMATH Keywords: small parameter
reviewed by: Vladimir Răsvan
MaRDI profile type: Publication
cites work: Q5848416
cites work: Q3254327
cites work: Q3925184
cites work: Q3931794
cites work: Q5516297
full work available at URL: https://doi.org/10.1007/bf01093179
OpenAlex ID: W2092203041
description: scientific article; zbMATH DE number 3842311