American Institute of Mathematical Sciences

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May  2013, 18(3): 797-819. doi: 10.3934/dcdsb.2013.18.797

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

Jie Tang 1, , Ziqing Xie 2, and  Zhimin Zhang 3,

1. 

College of Science, Hunan University of Technology, Zhuzhou, Hunan 412007, China
2. 

Key Laboratory of High Performance Computing and Stochastic Information Processing, College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China
3. 

Department of Mathematics, Wayne State University, Detroit, MI 48202

Received  October 2011 Revised  September 2012 Published  December 2012

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: spectral collocation method, Pantograph delay differential equations, vanishing proportional delay., domain decomposition, exponential convergence.
Mathematics Subject Classification: 34K28, 65L05, 65Q10, 41A1.
Citation: Jie Tang, Ziqing Xie, Zhimin Zhang. The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 797-819. doi: 10.3934/dcdsb.2013.18.797
References:
[1]

I. Ali, H. Brunner and T. Tang, A spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput. Math., 27 (2009), 254-265.  Google Scholar

[2]

I. Ali, H. Brunner and T. Tang, Spectral methods for pantograph-type differential and integral equations with multiple delays, Front. Math. China, 4 (2009), 49-61. doi: 10.1007/s11464-009-0010-z.  Google Scholar

[3]

A. Bellen, Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay, IMA J. Numer. Anal., 22 (2002), 529-536. doi: 10.1093/imanum/22.4.529.  Google Scholar

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A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar

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A. Bellen, H. Brunner, S. Maset and L. Torelli, Superconvergence in collocation methods on quasi-graded meshes for functional differential equations with vanishing delays, BIT, 46 (2006), 229-247. doi: 10.1007/s10543-006-0055-2.  Google Scholar

[6]

H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Differential Equations," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar

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H. Brunner, Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays, Front. Math. China, 4 (2009), 3-22. doi: 10.1007/s11464-009-0001-0.  Google Scholar

[8]

H. Brunner, Q. M. Huang and H. H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM J. Numer. Anal., 48 (2010), 1944-1967. doi: 10.1137/090771922.  Google Scholar

[9]

H. Brunner and Q. Y. Hu, Optimal superconvergence results for delay integro-differential equations of pantograph type, SIAM J. Numer. Anal., 45 (2007), 986-1004. doi: 10.1137/060660357.  Google Scholar

[10]

L. Fox, D. F. Mayers, J. R. Ockendon and A. B. Tayler, On a functional differential equation, J. Inst. Math. Appl., 8 (1971), 271-307.  Google Scholar

[11]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, "Spectral Methods. Fundamentals in Single Domains," Springer-Verlag, Berlin, 2006.  Google Scholar

[12]

B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comp. Math., 30 (2009), 249-280. doi: 10.1007/s10444-008-9067-6.  Google Scholar

[13]

B. Y. Guo and J. P. Yan, Legendre-Gauss collocation methods for initial value problems of second ordinary differential equations, App. Numer. Math., 59 (2009), 1386-1408. doi: 10.1016/j.apnum.2008.08.007.  Google Scholar

[14]

A. Iserles, On the generalized pantograph functional differential equation, Europ J. Appl. Math., 4 (1993), 1-38. doi: 10.1017/S0956792500000966.  Google Scholar

[15]

A. Iserles, On nonlinear delay-differential equations, Trans. Amer. Math. Soc., 344 (1994), 441-477. doi: 10.2307/2154725.  Google Scholar

[16]

T. Kato and J. B. Mcleod, The functional-differential equation $y'(x)=ay(\lambda x)+by(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937.  Google Scholar

[17]

T. Tang, X. Xu, and J. Cheng, On spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math., 26 (2008), 825-837.  Google Scholar

[18]

X. Tao, Z. Q. Xie, and X. J. Zhou, Spectral Petrov-Galerkin methods for the second kind Volterra type integro-differential equations, Numer. Math. Theor. Meth. Appl., 4 (2011), 216-236.  Google Scholar

[19]

Z. Q. Wang and L. L. Wang, A Legendre-Gauss collocation method for nonlinear delay differential equations, Dis. Cont. Dyn. Sys. B., 13 (2010), 685-708. doi: 10.3934/dcdsb.2010.13.685.  Google Scholar

show all references

References:
[1]

I. Ali, H. Brunner and T. Tang, A spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput. Math., 27 (2009), 254-265.  Google Scholar

[2]

I. Ali, H. Brunner and T. Tang, Spectral methods for pantograph-type differential and integral equations with multiple delays, Front. Math. China, 4 (2009), 49-61. doi: 10.1007/s11464-009-0010-z.  Google Scholar

[3]

A. Bellen, Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay, IMA J. Numer. Anal., 22 (2002), 529-536. doi: 10.1093/imanum/22.4.529.  Google Scholar

[4]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar

[5]

A. Bellen, H. Brunner, S. Maset and L. Torelli, Superconvergence in collocation methods on quasi-graded meshes for functional differential equations with vanishing delays, BIT, 46 (2006), 229-247. doi: 10.1007/s10543-006-0055-2.  Google Scholar

[6]

H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Differential Equations," Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.  Google Scholar

[7]

H. Brunner, Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays, Front. Math. China, 4 (2009), 3-22. doi: 10.1007/s11464-009-0001-0.  Google Scholar

[8]

H. Brunner, Q. M. Huang and H. H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM J. Numer. Anal., 48 (2010), 1944-1967. doi: 10.1137/090771922.  Google Scholar

[9]

H. Brunner and Q. Y. Hu, Optimal superconvergence results for delay integro-differential equations of pantograph type, SIAM J. Numer. Anal., 45 (2007), 986-1004. doi: 10.1137/060660357.  Google Scholar

[10]

L. Fox, D. F. Mayers, J. R. Ockendon and A. B. Tayler, On a functional differential equation, J. Inst. Math. Appl., 8 (1971), 271-307.  Google Scholar

[11]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, "Spectral Methods. Fundamentals in Single Domains," Springer-Verlag, Berlin, 2006.  Google Scholar

[12]

B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comp. Math., 30 (2009), 249-280. doi: 10.1007/s10444-008-9067-6.  Google Scholar

[13]

B. Y. Guo and J. P. Yan, Legendre-Gauss collocation methods for initial value problems of second ordinary differential equations, App. Numer. Math., 59 (2009), 1386-1408. doi: 10.1016/j.apnum.2008.08.007.  Google Scholar

[14]

A. Iserles, On the generalized pantograph functional differential equation, Europ J. Appl. Math., 4 (1993), 1-38. doi: 10.1017/S0956792500000966.  Google Scholar

[15]

A. Iserles, On nonlinear delay-differential equations, Trans. Amer. Math. Soc., 344 (1994), 441-477. doi: 10.2307/2154725.  Google Scholar

[16]

T. Kato and J. B. Mcleod, The functional-differential equation $y'(x)=ay(\lambda x)+by(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937.  Google Scholar

[17]

T. Tang, X. Xu, and J. Cheng, On spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math., 26 (2008), 825-837.  Google Scholar

[18]

X. Tao, Z. Q. Xie, and X. J. Zhou, Spectral Petrov-Galerkin methods for the second kind Volterra type integro-differential equations, Numer. Math. Theor. Meth. Appl., 4 (2011), 216-236.  Google Scholar

[19]

Z. Q. Wang and L. L. Wang, A Legendre-Gauss collocation method for nonlinear delay differential equations, Dis. Cont. Dyn. Sys. B., 13 (2010), 685-708. doi: 10.3934/dcdsb.2010.13.685.  Google Scholar

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