The long time behavior of a spectral collocation method for delay differential equations of pantograph type
type: Item
instance of: scholarly article
title: The long time behavior of a spectral collocation method for delay differential equations of pantograph type
zbMATH Open document ID: 1271.34077
author: Jie Tang
author: Zhimin Zhang
author: Ziqing Xie
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2013-04-24T00:00:00Z
review text: The authors present a spectral collocation method with domain decomposition (with a uniform mesh) for the pantograph-type delay differential equation of the form \[ y'(x)=a(x)y(x)+b(x)y(qx)+f(x),\;x \in [0,T],\;y(0)=y_0. \] An overview of existing numerical methods for solving this equation, along with comments relating to related problems and issues with their application, is given in Section 1. The authors comment on the increasing popularity of using a spectral method with partial differential equations and justify their decision to develop a ``novel `time stepping' scheme based on uniform meshes for large \(T\)'', stating that it has some ``remarkable advantages'' over Runge-Kutta methods and that it is more appropriate for predicting the long time behaviour of a dynamical system. In Section 2, the spectral collocation method is introduced and the existence and uniqueness of the spectral collocation solution is discussed. The numerical approach is derived using Legendre-Gauss interpolation and the numerical implementation of the collocation scheme is described. The discrete scheme is rewritten in matrix form as \(DU_m^N\), \(1\leq m \leq M\), where \(U_m^N(t)\) is the global solution to the pantograph-type equation. The structure of the resulting \(M\) systems depends strongly on the delay term in the spectral collocation equation, and changes for each value of \(m\) as the three phases, I, II and III (\(m=1, 1<m \leq q^{II}:= \lfloor{\frac{1}{1-q}}\rfloor\) and \(m \geq q^{III}:=\lceil{\frac{1}{1-q}}\rceil\)). In the third phase, the pure delay phase, it is necessary to deal with two cases separately. A brief discussion of the existence and uniqueness of the spectral collocation solution, which is defined by a linear algebraic system, is included. A theorem is presented and proved. Convergence analysis is the focus of Section 3. Exponential convergence of the global errors is proved, both theoretically and numerically, and error estimates are proved. The main r…
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34K07
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L60
zbMATH DE Number: 6157330
zbMATH Keywords: vanishing proportional delay
zbMATH Keywords: exponential convergence
zbMATH Keywords: pantograph delay differential equations
zbMATH Keywords: spectral collocation method
zbMATH Keywords: domain decomposition
reviewed by: Pat Lumb
MaRDI profile type: Publication
full work available at URL: dcdsb.2013.18.797
OpenAlex ID: W2031582911
DOI: 10.3934/DCDSB.2013.18.797
description: scientific article; zbMATH DE number 6157330