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 and 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

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