Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one
type: Item
instance of: scholarly article
title: Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one
zbMATH Open document ID: 0859.34039
author: T. M. Dunster
published in: Methods and Applications of Analysis
publication date: 1996-12-03T00:00:00Z
review text: The author investigates solutions of the general homogeneous linear second order differential equation of the form \[ {{d^2w}\over{dz^2}}+ f(z){{dw}\over{dz}}+ g(z)w=0. \tag{\(*\)} \] In \S 2, the author gives error bounds which are uniformly valid for \(0\leq|\text{arg}(ze^{-\pi(j- 1)i})|\leq 2\pi\). In \S 3, he gives details of the proof of the derivation of these bounds, which uses the technique of successive approximations. In \S 4, the author generalizes the results of \S\S 2-3 to give exponentially improved expansions with an improved relative error term of \(O(z^{-m})\) as \(z\to\infty\) where \(m\) is a prescribed fixed positive integer. In \S 5, he gives brief details on the extension of the error analysis to sectors, in conjunction with the results of \S 4. In \S 6, the author examines in more detail the asymptotic nature of the error bounds, which involve so-called weight functions. Finally, in \S 7, he gives a numerical example on the calculation of certain constants which appear.
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 30E15
zbMATH DE Number: 935323
zbMATH Keywords: homogeneous linear second order differential equation of the form
reviewed by: Ion Badea
MaRDI profile type: Publication
DOI: 10.4310/MAA.1996.V3.N1.A7
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description: scientific article; zbMATH DE number 935323