Recent advances in the numerical analysis of Volterra functional differential equations with variable delays
type: Item
instance of: scholarly article
title: Recent advances in the numerical analysis of Volterra functional differential equations with variable delays
zbMATH Open document ID: 1170.65103
author: H. Brunner
published in: Journal of Computational and Applied Mathematics
publication date: 2009-06-04T00:00:00Z
review text: The author considers initial value problems of a linear (ordinary) delay differential equation involving Volterra integral operators, i.e., a Volterra functional integro-differential equation (VFIDE). Thereby, the delay is a predetermined nonlinear time-dependent function, which may be equal to zero at the beginning (vanishing delay). Omitting the integral operators, the linear delay differential equation reduces to the pantograph equation, [cf. \textit{M. Arnold} and \textit{B. Simeon} [Appl. Numer. Math. 34, 345--362 (2000; Zbl 0964.65101)]. Considering the VFIDE, a collocation method is constructed straightforward using piecewise polynomials of degree not exceeding \(m\) on a uniform mesh. In case of non-vanishing delay, previous work shows an optimal global (uniform) superconvergence of order \(m+1\), whereas the optimal local superconvergence exhibits the order \(2m\) in the collocation points. In case of vanishing delay, the author proves an optimal global (uniform) convergence of order \(m\) and an optimal local superconvergence of order \(m+2\) for \(m \geq 3\). The generalisation of the results to a VFIDE of second order is outlined. Finally, the author specifies some open problems in this context, for example, the asymptotic behaviour of the exact solutions or of the numerical solutions for VFIDEs. Numerical results of test examples are not within the scope of the paper. Since the author wants to provide a survey on recent results, a huge number of references is given.
reviewed by: Roland Pulch
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 34K06
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5561630
zbMATH Keywords: Volterra functional integro-differential equations
zbMATH Keywords: delay differential equations
zbMATH Keywords: vanishing delays
zbMATH Keywords: Volterra integral operator
zbMATH Keywords: pantograph equation
zbMATH Keywords: collocation method
zbMATH Keywords: superconvergence
zbMATH Keywords: error analysis
zbMATH Keywords: uniform mesh
MaRDI profile type: Publication
full work available at URL: j.cam.2008.03.024
OpenAlex ID: W1995908585
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Recent advances in the numerical analysis of Volterra functional differential equations with variable delays (DOI)
DOI: 10.1016/J.CAM.2008.03.024

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