An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system
type: Item
instance of: scholarly article
title: An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system
zbMATH Open document ID: 0867.65034
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 1997-07-13T00:00:00Z
review text: This paper deals with the solution of a first-order linear system of differential equations with variable coefficients around a singularity of their coefficients. More precisely, the linear systems under consideration have the form \(x{\mathbf y}' =A(x) {\mathbf y}\), where \(A(x)= x^{-q} \sum_{j\geq 0} A_jx^j\) with an integer \(q\geq 0\) and \(A_j\) constant matrices, and the aim of the paper is to compute the exponential part of a fundamental matrix solution around the singularity \(x=0\). It is known [cf. \textit{W. Wasow}: Asymptotic expansions for ordinary differential equations (1965; Zbl 0133.35301)] that, after a meromorphic transformation, there is an integer \(s\geq 1\) -- the so-called ramification index -- such that a fundamental matrix of these systems can be written as \(Y(x)= \Phi(u)x^R \exp (Q(u))\), where \(u= x^{(-1/s)}\), \(\Phi(u)\) is a meromorphic series in \(u\), \(R\) a constant matrix and \(Q(u)= \text{diag} (q_1(u), \dots, q_n(u))\). In this context the author proposes an efficient algorithm to compute \(Q(u)\). Furthermore he gives two versions of this algorithm that can be implemented in the Maple computer algebra system and presents some examples to show the efficiency of the proposed algorithm.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 978144
zbMATH Keywords: singular points
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: first-order linear system
zbMATH Keywords: variable coefficients
zbMATH Keywords: ramification index
zbMATH Keywords: fundamental matrix
zbMATH Keywords: algorithm
zbMATH Keywords: Maple computer algebra system
author: Moulay A. Barkatou
reviewed by: Manuel Calvo
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: s002000050048
OpenAlex ID: W1988098065
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DOI: 10.1007/S002000050048
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description: scientific article; zbMATH DE number 978144