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

Eric A. Butcher, Oleg A. Bobrenkov

Research output: Contribution to journalArticle

73 Scopus citations

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

Keywords

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

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics

Fingerprint Dive into the research topics of 'On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations'. Together they form a unique fingerprint.

  • Cite this