Dissipativity of \(\theta \)-methods for nonlinear delay differential equations of neutral type
type: Item
instance of: scholarly article
title: Dissipativity of \(\theta \)-methods for nonlinear delay differential equations of neutral type
zbMATH Open document ID: 1166.65036
published in: Applied Numerical Mathematics
publication date: 2009-04-30T00:00:00Z
review text: This paper is concerned with the dissipativity properties of nonlinear delay differential equations of neutral type with the form \[ y'(t) = F (t, y(t), G(t, y(t-\tau ), y'(t- \tau ))),\quad t \geq 0\tag{1} \] where \( \tau \) is a positive constant, \( y: \mathbb{R} \to H \) (real or complex Hilbert space) and \( y(t) = g(t)\), \(t\leq 0\) is the given initial function. Here the evolutionary functional equation (1) is said to be dissipative in \(H\) if there is an absorbing bounded set \( B \subset H \) such that for all bounded set \(D\) and initial function \( g(t) \in D\) there is \( t_0 = t_0(D)\) such that the corresponding solution of (1) satisfies \( y(t) \in B \) for all \( t \geq t_0\). In this context the author gives a set of sufficient conditions on functions \(F\) and \(G\) that ensure the dissipativity in the above mentioned context. These sufficient conditions extend, in some sense, the ideas of \textit{A. M. Stuart} and \textit{A. R. Humphries} [SIAM Rev. 36, No.~2, 226--257 (1994; Zbl 0807.65091)] for standard ordinary differential equations to nonlinear delay differential equations of neutral type. In the second part of the paper the dissipativity of linear \(\theta\)-methods in finite dimensional spaces is studied. Now under the theoretical sufficient conditions of dissipativity it is proved that the above \(\theta\)--methods are dissipative for \( 1/2 < \theta \leq 1\). The paper ends with an interesting 2D example that confirms the previous analysis and also shows that the conditions of the above analysis are only sufficient but not necessary for dissipativity.
reviewed by: Manuel Calvo
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34K12
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 34K40
zbMATH DE Number: 5550384
zbMATH Keywords: neutral delay differential equations
zbMATH Keywords: dissipativity of solutions
zbMATH Keywords: \(\theta\)-methods
zbMATH Keywords: numerical examples
author: Siqing Gan
MaRDI profile type: Publication
full work available at URL: j.apnum.2008.08.003
OpenAlex ID: W2008586840
cites work: On the contractivity and asymptotic stability of systems of delay differential equations of neutral type
cites work: Numerical stability of nonlinear delay differential equations of neutral type
cites work: Non-linear stability of a general class of differential equation methods
cites work: Q4122713
cites work: Dissipativity of linear \(\theta \)-methods for integro-differential equations
cites work: Dissipativity of \(\theta\)-methods for nonlinear Volterra delay-integro-differential equations
cites work: Q5432718
cites work: Global Dissipativity for A-Stable Methods
cites work: Dissipativity of Runge-Kutta methods in Hilbert spaces
cites work: Stability analysis of numerical methods for systems of neutral delay-differential equations
cites work: Dissipativity of Runge-Kutta methods for dynamical systems with delays
cites work: Dissipativity of one-leg methods for dynamical systems with delays
cites work: Linear stability of general linear methods for systems of neutral delay differential equations
cites work: Model Problems in Numerical Stability Theory for Initial Value Problems
cites work: Runge–Kutta Methods for Dissipative and Gradient Dynamical Systems
cites work: Numerical Solution of Implicit Neutral Functional Differential Equations
cites work: The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations
cites work: Q4333039
cites work: NUMERICAL AND ANALYTIC DISSIPATIVITY OF THE θ-METHOD FOR DELAY DIFFERENTIAL EQUATIONS WITH A BOUNDED VARIABLE LAG
cites work: A stable numerical approach for implicit non-linear neutral delay differential equations
cites work: Q5866785
cites work: Q5866937
cites work: Q4795619
cites work: The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations
DOI: 10.1016/J.APNUM.2008.08.003
description: scientific article; zbMATH DE number 5550384