On the asymptotic behavior of solutions of a certain second order ordinary differential equation
type: Item
instance of: scholarly article
title: On the asymptotic behavior of solutions of a certain second order ordinary differential equation
zbMATH Open document ID: 0606.34036
author: Atsushi Yoshikawa
published in: Kodai Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: A fundamental pair of solutions is given for the differential equation \(u_{tt}+k^ 2(t^ 2+y^ 2)u+kc=0,\) where u(t,y) is the unknown function, \(k>0\), c is a complex constant and (t,y) is in \(R^ 2\). The novelty lies in its derivation and the corresponding asymptotic expansion, which is different than the given usually. Let \(\epsilon\) and \(\delta\) be the symbols \(+\) or -. Then, the fundamental theorem states that for each \(\epsilon \in \{+,-\}\) there is a pair of fundamental solutions \(v^{\delta}_{\epsilon}(t,y,k)\), defined and infinitely differentiable in some domain \(D_{\epsilon}\) of the (t,y,k) space, and their asymptotic expansions are given (they are related to confluent hypergeometric functions). The given results are expected to generalize to solutions of the partial differential equation \(u_{tt}- u_{xx}+(t^ 2+x^ 2+y^ 2)k^ 2+kc=0.\)
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 3979563
zbMATH Keywords: second order differential operator
zbMATH Keywords: asymptotic expansion
Wikidata QID: Q115224627
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037103
OpenAlex ID: W2049760612
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DOI: 10.2996/KMJ/1138037103
description: scientific article; zbMATH DE number 3979563