Oscillation analysis of numerical solution in the \(\theta\)-methods for equation \(x\prime (t) + ax(t) + a_{1}x([t - 1]) = 0\)
type: Item
instance of: scholarly article
title: Oscillation analysis of numerical solution in the \(\theta\)-methods for equation \(x\prime (t) + ax(t) + a_{1}x([t - 1]) = 0\)
zbMATH Open document ID: 1118.65080
author: Jianfang Gao
author: M. Z. Liu
author: Z. W. Yang
published in: Applied Mathematics and Computation
publication date: 2007-04-26T00:00:00Z
review text: This paper is concerned with the preservation of the oscillatory behavior of the solutions of functional differential equations (1) \( x'(t) + a x(t) + a_1 x([t-1])=0\), \(t>0\), where \( a, a_1 >0\) are constants and \([\cdot]\) denotes the greatest integer functions, when integrated numerically by the \(\theta\)-methods. Since for any given set of starting values \( x(-1), x(0)\) there exist a unique solution of (1) which is oscillatory iff \( 4 a_1 (1-e^{-a})-a>0\), some conditions on the \(\theta\)--methods are given for the preservation of the oscillatory behavior of the solutions of (1) for sufficiently small step size. Further it is shown that the zeros of the linear interpolation of the numerical solutions converge to the zeros of the exact solution with first order accuracy when \( \theta \neq 1/2\) and second order accuracy when \( \theta =1/2\). Finally the results of some numerical experiments are presented to show the oscillatory behavior preservation according to the values of the step size.
reviewed by: Manuel Calvo
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34K06
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34K28
zbMATH DE Number: 5147034
zbMATH Keywords: linear delay differential equations
zbMATH Keywords: initial value problems
zbMATH Keywords: preservation of oscillatory behavior
zbMATH Keywords: Theta methods
zbMATH Keywords: functional differential equations
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: j.amc.2006.07.119
OpenAlex ID: W2061588896
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DOI: 10.1016/J.AMC.2006.07.119
description: scientific article; zbMATH DE number 5147034