Spectral methods for pantograph-type differential and integral equations with multiple delays

doi:10.1007/s11464-009-0010-z

Front Math Chin

2009, Vol. 4

Issue (1) : 49-61 https://doi.org/10.1007/s11464-009-0010-z

RESEARCH ARTICLE

Spectral methods for pantograph-type differential and integral equations with multiple delays

Ishtiaq ALI^1,2(

), Hermann BRUNNER^3,4, Tao TANG⁴

1. Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China; 2. Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan; 3. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada; 4. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China

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Abstract

We analyze the convergence properties of the spectral method when used to approximate smooth solutions of delay differential or integral equations with two or more vanishing delays. It is shown that for the pantograph-type functional equations the spectral methods yield the familiar exponential order of convergence. Various numerical examples are used to illustrate these results.

Keywords Delay differential equation Volterra functional integral equation multiple vanishing delays Legendre spectral method convergence analysis

Corresponding Author(s): ALI Ishtiaq,Email:ishtiaq@lsec.cc.ac.cn, ishtiaqali@comsats.edu.pk

Issue Date: 05 March 2009

Cite this article:

Ishtiaq ALI,Hermann BRUNNER,Tao TANG. Spectral methods for pantograph-type differential and integral equations with multiple delays[J]. Front Math Chin, 2009, 4(1): 49-61.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-009-0010-z
https://academic.hep.com.cn/fmc/EN/Y2009/V4/I1/49

1	Ali I, Brunner H, Tang T. A spectral method for pantograph-type delay differential equations and its convergence analysis
2	Bellen A, Zennaro M. Numerical Methods for Delay Differentials Equations. Oxford: Oxford University Press, 2003 doi: 10.1093/acprof:oso/9780198506546.001.0001
3	Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge: Cambridge University Press, 2004
4	Canuto C, Hussaini M, Quarteroni A, Zang T A. Spectral Methods: Fundamentals in Single Domains. Berlin: Springer, 2006
5	Derfel G A, Vogl F. On the asymptotics of solutions of a class of linear functionaldifferential equations. Europ J Appl Math , 1996, 7: 511-518 doi: 10.1017/S0956792500002527
6	Guglielmi N. Short proofs and a counterexample for analytical and numerical stability of delay equations with infinite memory. IMA J Numer Anal , 2006, 26: 60-77 doi: 10.1093/imanum/dri021
7	Guglielmi N, Hairer E. Implementing Radau IIA methods for stiff delay differential equations. Computing , 2001, 67: 1-12 doi: 10.1007/s006070170013
8	Iserles A. On the generalized pantograph functional-differential equation. Europ J Appl Math , 1993, 4: 1-38 doi: 10.1017/S0956792500000966
9	Iserles A, Liu Y-K. On pantograph integro-differential equations. J Integral Equations Appl , 1994, 6: 213-237 doi: 10.1216/jiea/1181075805
10	Ishiwata E. On the attainable order of collocation methods for the neutral functionaldifferential equations with proportional delays. Computing , 2000, 64: 207-222 doi: 10.1007/s006070050044
11	Li D, Liu M Z. Asymptotic stability of numerical solution of pantograph delay differential equations. J Harbin Inst Tech , 1999, 31: 57-59 (in Chinese)
12	Liu M Z, Li D. Properties of analytic solution and numerical solution of multipantograph equation. Appl Math Comput , 2004, 155: 853-871 doi: 10.1016/j.amc.2003.07.017
13	Mastroianni G, Occorsio D. Optimal systems of nodes for Lagrange interpolation on bounded intervals: A survey. J Comput Appl Math , 2001, 134: 325-341 doi: 10.1016/S0377-0427(00)00557-4
14	Qiu L, Mitsui T, Kuang J X. The numerical stability of the θ-method for delay differential equations with many variable delays. J Comput Math , 1999, 17: 523-532
15	Shen J, Tang T. Spectral and High-Order Methods with Applications. Beijing: Science Press, 2006
16	Tang T, Xu X, Cheng J. On spectral methods for Volterra type integral equations and the convergence analysis. J Comput Math , 2008, 26: 825-837
17	Zhao J, Xu Y, Qiao Y. The attainable order of the collocation method for doublepantograph delay differential equation. Numer Math J Chinese Univ , 2005, 27: 297-308 (in Chinese)

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