A stable numerical approach for implicit non-linear neutral delay differential equations
type: Item
instance of: scholarly article
title: A stable numerical approach for implicit non-linear neutral delay differential equations
zbMATH Open document ID: 1030.65078
DOI: 10.1023/A:1023613425081
published in: BIT
publication date: 2003-08-06T00:00:00Z
review text: The paper considers implicit neutral functional differential equations or functional differential equations n Hale's form: \[ (y(t)-G(t,y(t-\tau(t))))' = F(t,y(t),y(t-\tau(t))), \quad t \geq t_0 \] under suitable conditions on \(F,G\) which ensure the existence of a unique solution subject to the specification of a suitable initial function. After preliminaries, the authors provide a reformulation of the equation as a delay differential algebraic equation and use the reformulated problem as the basis for an analysis of boundedness and stability of the zero solution under small perturbations in the initial function. Although the analysis is given for the nonlinear problem, careful attention is paid to the important linear case. Numerical schemes for solution of the problem are considered, based on a continuous Runge-Kutta method, and the authors give conditions under which the stability of the underlying problem is preserved in the numerical approximation. These results are backed up by the use of examples in which they are able to illustrate the stability properties of the Euler method and of the 2-stage Lobatto IIIC method.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 65L80
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34A09
zbMATH DE Number: 1960119
zbMATH Keywords: Neutral delay differential equation
zbMATH Keywords: Runge Kutta method
zbMATH Keywords: stability analysis
zbMATH Keywords: numerical examples
zbMATH Keywords: implicit neutral functional differential equations
zbMATH Keywords: delay differential algebraic equation
zbMATH Keywords: Euler method
zbMATH Keywords: 2-stage Lobatto IIIC method
author: R. Vermiglio
author: Lucio Torelli
reviewed by: Neville J. Ford
MaRDI profile type: Publication
full work available at URL: a:1023613425081
OpenAlex ID: W29277913
description: scientific article; zbMATH DE number 1960119