The long time behavior of a spectral collocation method for delay differentialequationsof pantograph type

Jie Tang; Ziqing Xie; Zhimin Zhang

doi:10.3934/dcdsb.2013.18.797

Article Contents

2013, Volume 18, Issue 3: 797-819. Doi: 10.3934/dcdsb.2013.18.797

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The long time behavior of a spectral collocation method for delay differential equations of pantograph type

1.
College of Science, Hunan University of Technology, Zhuzhou, Hunan 412007
2.
Key Laboratory of High Performance Computing and Stochastic Information Processing, College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081
3.
Department of Mathematics, Wayne State University, Detroit, MI 48202

Received: September 30, 2011

Revised: August 31, 2012

Published: April 2013

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Abstract

In this paper, we propose an efficient numerical method for delay differential equations with vanishing proportional delay qt (0 < q < 1). The algorithm is a mixture of the Legendre-Gauss collocation method and domain decomposition. It has global convergence and spectral accuracy provided that the data in the given pantograph delay differential equation are sufficiently smooth. Numerical results demonstrate the spectral accuracy of this approach and coincide well with theoretical analysis.

Keywords:

Pantograph delay differential equations,
spectral collocation method,
exponential convergence,
domain decomposition,
vanishing proportional delay.

Mathematics Subject Classification: 34K28, 65L05, 65Q10, 41A10.

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