Delay-dependent stability analysis of trapezium rule for second order delay differential equations with three parameters
type: Item
instance of: scholarly article
title: Delay-dependent stability analysis of trapezium rule for second order delay differential equations with three parameters
zbMATH Open document ID: 1207.65102
published in: Journal of the Franklin Institute
publication date: 2010-12-20T00:00:00Z
review text: The paper is concerned with the study of the conservation of stability properties by applying the trapezoidal rule to the second order delay differential equations with three parameters: \[ y''=ay'(t)+by(t)+cy(t-\tau), \quad t\geq 0. \] This problem which in general cannot be solved exactly is replaced by using the trapezoidal rule by the numerical method \[ \begin{aligned} y_{n+1}-y_{n}&=\tfrac{1}{2}\, hY_{n+1}+\tfrac{1}{2}\, hY_{n},\\ Y_{n+1}-Y_{n}&=\tfrac{1}{2}\, haY_{n+1}+\tfrac{1}{2}\, haY_{n}+\tfrac{1}{2}\, hby_{n+1}+\tfrac{1}{2}\, hby_{n}+\tfrac{1}{2}\, hcy_{n+1-m}+\tfrac{1}{2}\,hc y_{n-m},\end{aligned} \] where \(h=\tau /m\) is a consistent step size and \(Y_{n}\) an approximation to \(y(x_{n})\). The main result is given in the following theorem: The difference equation is asymptotically stable if: \[ a<-b\tau \quad \text{and}\quad b\tau^{2}<-c\tau^{}<\frac{4m^{2}(1-\cos\eta)+h^{2}b(1+\cos\eta)}{\cos m\eta(1+\cos\eta)} \] where \(\eta\) is the root of \[ a=-((4m(1-\cos\eta)+h^{2}b(1+\cos\eta ))/(2\tau\sin\eta)\tan m\eta \] with \(\eta\in(0,\pi/2m)\).
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 65L03
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 34K28
zbMATH DE Number: 5828189
zbMATH Keywords: Delay dependent stability
zbMATH Keywords: trapezoidal rule
zbMATH Keywords: conservation of stability
zbMATH Keywords: second order delay differential equations
author: Wenhao Li
author: Chengming Huang
author: Siqing Gan
MaRDI profile type: Publication
full work available at URL: j.jfranklin.2010.06.013
OpenAlex ID: W1988734128
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DOI: 10.1016/J.JFRANKLIN.2010.06.013
description: scientific article; zbMATH DE number 5828189