American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685
[2]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
[3]

Qiumei Huang, Xiuxiu Xu, Hermann Brunner. Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5423-5443. doi: 10.3934/dcds.2016039
[4]

Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367
[5]

C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002
[6]

Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
[7]

Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029
[8]

Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064
[9]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[10]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[11]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
[12]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[13]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098
[14]

Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092
[15]

Eugenii Shustin. Exponential decay of oscillations in a multidimensional delay differential system. Conference Publications, 2003, 2003 (Special) : 809-816. doi: 10.3934/proc.2003.2003.809
[16]

Michael Dellnitz, Mirko Hessel-Von Molo, Adrian Ziessler. On the computation of attractors for delay differential equations. Journal of Computational Dynamics, 2016, 3 (1) : 93-112. doi: 10.3934/jcd.2016005
[17]

Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751
[18]

Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827
[19]

Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257
[20]

Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences