On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations

Eric Butcher, Oleg A. Bobrenkov

Research output: Contribution to journalArticle

67 Citations (Scopus)

Abstract

In this paper, the approximation technique proposed in Breda et al. (2005) [1] for converting a linear system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is extended to linear and nonlinear systems of DDEs with time-periodic coefficients. The Chebyshev spectral continuous time approximation (ChSCTA) technique is used to study the stability of first and second-order constant coefficient DDEs, a delayed system with a cubic nonlinearity and parametric sinusoidal excitation, the delayed Mathieu's equation, and delayed systems with two fixed delays. In all the examples, the stability and time response obtained from ChSCTA show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. The " spectral accuracy" convergence behavior of Chebyshev spectral collocation shown in Trefethen (2000) [2] which the proposed technique possesses is compared to the convergence properties of finite difference-based continuous time approximation for constant-coefficient DDEs proposed recently in Sun (2009) [3] and Sun and Song (2009) [4].

Original languageEnglish (US)
Pages (from-to)1541-1554
Number of pages14
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume16
Issue number3
DOIs
StatePublished - Mar 2011
Externally publishedYes

Fingerprint

Delay Differential Equations
Chebyshev
Continuous Time
Differential equations
Approximation
Sun
Linear systems
Coefficient
Linear Systems
Mathieu Equation
Periodic Coefficients
Collocation
System of Ordinary Differential Equations
Approximation Methods
Ordinary differential equations
Convergence Properties
Response Time
Nonlinear systems
Finite Difference
Nonlinear Systems

Keywords

  • Chebyshev collocation
  • Continuous time approximation
  • Periodic delay differential equations

ASJC Scopus subject areas

  • Modeling and Simulation
  • Numerical Analysis
  • Applied Mathematics

Cite this

@article{7d3010e41997402daa2e967ea0c8e750,
title = "On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations",
abstract = "In this paper, the approximation technique proposed in Breda et al. (2005) [1] for converting a linear system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is extended to linear and nonlinear systems of DDEs with time-periodic coefficients. The Chebyshev spectral continuous time approximation (ChSCTA) technique is used to study the stability of first and second-order constant coefficient DDEs, a delayed system with a cubic nonlinearity and parametric sinusoidal excitation, the delayed Mathieu's equation, and delayed systems with two fixed delays. In all the examples, the stability and time response obtained from ChSCTA show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. The {"} spectral accuracy{"} convergence behavior of Chebyshev spectral collocation shown in Trefethen (2000) [2] which the proposed technique possesses is compared to the convergence properties of finite difference-based continuous time approximation for constant-coefficient DDEs proposed recently in Sun (2009) [3] and Sun and Song (2009) [4].",
keywords = "Chebyshev collocation, Continuous time approximation, Periodic delay differential equations",
author = "Eric Butcher and Bobrenkov, {Oleg A.}",
year = "2011",
month = "3",
doi = "10.1016/j.cnsns.2010.05.037",
language = "English (US)",
volume = "16",
pages = "1541--1554",
journal = "Communications in Nonlinear Science and Numerical Simulation",
issn = "1007-5704",
publisher = "Elsevier",
number = "3",

}

TY - JOUR

T1 - On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations

AU - Butcher, Eric

AU - Bobrenkov, Oleg A.

PY - 2011/3

Y1 - 2011/3

N2 - In this paper, the approximation technique proposed in Breda et al. (2005) [1] for converting a linear system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is extended to linear and nonlinear systems of DDEs with time-periodic coefficients. The Chebyshev spectral continuous time approximation (ChSCTA) technique is used to study the stability of first and second-order constant coefficient DDEs, a delayed system with a cubic nonlinearity and parametric sinusoidal excitation, the delayed Mathieu's equation, and delayed systems with two fixed delays. In all the examples, the stability and time response obtained from ChSCTA show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. The " spectral accuracy" convergence behavior of Chebyshev spectral collocation shown in Trefethen (2000) [2] which the proposed technique possesses is compared to the convergence properties of finite difference-based continuous time approximation for constant-coefficient DDEs proposed recently in Sun (2009) [3] and Sun and Song (2009) [4].

AB - In this paper, the approximation technique proposed in Breda et al. (2005) [1] for converting a linear system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is extended to linear and nonlinear systems of DDEs with time-periodic coefficients. The Chebyshev spectral continuous time approximation (ChSCTA) technique is used to study the stability of first and second-order constant coefficient DDEs, a delayed system with a cubic nonlinearity and parametric sinusoidal excitation, the delayed Mathieu's equation, and delayed systems with two fixed delays. In all the examples, the stability and time response obtained from ChSCTA show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. The " spectral accuracy" convergence behavior of Chebyshev spectral collocation shown in Trefethen (2000) [2] which the proposed technique possesses is compared to the convergence properties of finite difference-based continuous time approximation for constant-coefficient DDEs proposed recently in Sun (2009) [3] and Sun and Song (2009) [4].

KW - Chebyshev collocation

KW - Continuous time approximation

KW - Periodic delay differential equations

UR - http://www.scopus.com/inward/record.url?scp=77957355390&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957355390&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2010.05.037

DO - 10.1016/j.cnsns.2010.05.037

M3 - Article

AN - SCOPUS:77957355390

VL - 16

SP - 1541

EP - 1554

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

IS - 3

ER -