On an \(n\)th-order linear ordinary differential equation with a turning-singular point
type: Item
instance of: scholarly article
title: On an \(n\)th-order linear ordinary differential equation with a turning-singular point
zbMATH Open document ID: 0911.34053
author: Minoru Nakano
published in: Tokyo Journal of Mathematics
publication date: 1999-05-04T00:00:00Z
review text: The author considers the system \(\sum^n_{k=1} \varepsilon^{(n-k)h}P_k (x, \varepsilon)y^{(n-k)} =\varepsilon^{nh} y^{(n)}\), in a suitable identified space with favorable conditions imposed on the coefficients, with the asymptotic form \(P_k(x, \varepsilon)= p_k.x^{k m}+ O((\varepsilon x^{-1/ \alpha})^l)\) for all \(k\) with \(\alpha= 1/(m+r)>0\), and \(x\) in the region \(K\varepsilon^\alpha \leq| x|\leq x_0\), for some positive constant \(K\). Adopting a stretching change \(x=t \varepsilon^\alpha\), the above differential equation is transformed to the system: \(\varepsilon^{nh}y^{(n)}= \sum\varepsilon^{(n-k) h^.} p_k.P^k(t)y^{(n-k)}\), \(h'=h-(m+1)\), \(\alpha>0\). The author constructs solutions to the system according to the WKB method and uses the asymptotic expansion for the inner and outer solutions to prove that every solution to the original system converges uniformly. The topology of the local and global Stokes curves is discussed and brilliant graphs of these curves are represented. Many references are given related to the subject. The reviewer finds this whole article exposed in such an intriguing way that is very enjoyable to read.
reviewed by: Hussain S. Nur
Mathematics Subject Classification ID: 34E20
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34M60
zbMATH DE Number: 1207094
zbMATH Keywords: \(n\)th-order linear ordinary
zbMATH Keywords: differential equation
zbMATH Keywords: turning-singular point
Wikidata QID: Q115219823
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270041996
OpenAlex ID: W2015699702
DOI: 10.3836/TJM/1270041996
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description: scientific article; zbMATH DE number 1207094