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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, 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 |
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.
|
[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.
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.
doi: 10.1093/imanum/22.4.529. |
[4] |
A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations,", Oxford University Press, (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.
doi: 10.1007/s10543-006-0055-2. |
[6] |
H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Differential Equations,", Cambridge University Press, (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.
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.
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.
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.
|
[11] |
C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, "Spectral Methods. Fundamentals in Single Domains,", Springer-Verlag, (2006).
|
[12] |
B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations,, Adv. Comp. Math., 30 (2009), 249.
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.
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.
doi: 10.1017/S0956792500000966. |
[15] |
A. Iserles, On nonlinear delay-differential equations,, Trans. Amer. Math. Soc., 344 (1994), 441.
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.
|
[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.
|
[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.
|
[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.
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.
|
[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.
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.
doi: 10.1093/imanum/22.4.529. |
[4] |
A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations,", Oxford University Press, (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.
doi: 10.1007/s10543-006-0055-2. |
[6] |
H. Brunner, "Collocation Methods for Volterra Integral and Related Functional Differential Equations,", Cambridge University Press, (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.
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.
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.
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.
|
[11] |
C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, "Spectral Methods. Fundamentals in Single Domains,", Springer-Verlag, (2006).
|
[12] |
B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations,, Adv. Comp. Math., 30 (2009), 249.
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.
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.
doi: 10.1017/S0956792500000966. |
[15] |
A. Iserles, On nonlinear delay-differential equations,, Trans. Amer. Math. Soc., 344 (1994), 441.
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.
|
[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.
|
[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.
|
[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.
doi: 10.3934/dcdsb.2010.13.685. |
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