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Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type
1. | Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China, China |
2. | Department of Mathematics, Hong Kong Baptist University, Hong Kong |
References:
[1] |
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. |
[2] |
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 Numer. Math., 46 (2006), 229-247.
doi: 10.1007/s10543-006-0055-2. |
[3] |
A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\theta$-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 279-293.
doi: 10.1016/S0168-9274(97)00026-3. |
[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] |
H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511543234. |
[6] |
H. Brunner, Q. Hu and Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal., 21 (2001), 783-798.
doi: 10.1093/imanum/21.4.783. |
[7] |
H. Brunner, Q. Huang and H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM. J. Numer. Anal., 48 (2010), 1944-1967.
doi: 10.1137/090771922. |
[8] |
H. Brunner and H. Liang, Stability of collocation methods for delay differential equations with vanishing delays, BIT Numer. Math., 50 (2010), 693-711.
doi: 10.1007/s10543-010-0285-1. |
[9] |
C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Press of Science and Technology, Changsha, 2001 (in Chinese). |
[10] |
M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp., 36 (1981), 455-473.
doi: 10.1090/S0025-5718-1981-0606506-0. |
[11] |
K. Deng, Z. Xiong and Y. Huang, The Galerkin continuous finite element method for delay-differential equation with a variable term, Appl. Math. Comput., 186 (2007), 1488-1496.
doi: 10.1016/j.amc.2006.07.147. |
[12] |
Q. Huang, H. Xie and H. Brunner, Superconvergence of discontinuous Galerkin solutions for delay differential equation of pantograph type, SIAM. J. Sci. Comput., 33 (2011), 2664-2684.
doi: 10.1137/110824632. |
[13] |
A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38.
doi: 10.1017/S0956792500000966. |
[14] |
T. Koto, Stability of Runge-Kutta methods for the generalized pantograph equation, Numer. Math., 84 (1999), 233-247.
doi: 10.1007/s002110050470. |
[15] |
D. Li and C. Zhang, Superconvergence of a discontinuous Galerkin Method for first-order linear delay differential equations, J. Comput. Math., 29 (2011), 574-588.
doi: 10.4208/jcm.1107-m3433. |
[16] |
Y. Liu, On the $\theta$ -method for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190.
doi: 10.1016/0377-0427(95)00222-7. |
[17] |
N. Takama, Y. Muroya and E. Ishiwata, On the attainable order of collocation methods for delay differential equations with proportional delay, BIT Numer. Math., 40 (2000), 374-394.
doi: 10.1023/A:1022351309662. |
[18] |
X. Xu and Q. Huang, Continuous Galerkin methods for delay differential equations of pantograph type, Math. Pract. Theory, 24 (2014), 280-288 (in Chinese). |
[19] |
X. Xu, Q. Huang and H. Chen, Local superconvergence of continuous Galerkin solutions for delay differential equations of pantograph type, J. Comput. Math., 34 (2016), 186-199. |
show all references
References:
[1] |
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. |
[2] |
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 Numer. Math., 46 (2006), 229-247.
doi: 10.1007/s10543-006-0055-2. |
[3] |
A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\theta$-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 279-293.
doi: 10.1016/S0168-9274(97)00026-3. |
[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] |
H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511543234. |
[6] |
H. Brunner, Q. Hu and Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal., 21 (2001), 783-798.
doi: 10.1093/imanum/21.4.783. |
[7] |
H. Brunner, Q. Huang and H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM. J. Numer. Anal., 48 (2010), 1944-1967.
doi: 10.1137/090771922. |
[8] |
H. Brunner and H. Liang, Stability of collocation methods for delay differential equations with vanishing delays, BIT Numer. Math., 50 (2010), 693-711.
doi: 10.1007/s10543-010-0285-1. |
[9] |
C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Press of Science and Technology, Changsha, 2001 (in Chinese). |
[10] |
M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp., 36 (1981), 455-473.
doi: 10.1090/S0025-5718-1981-0606506-0. |
[11] |
K. Deng, Z. Xiong and Y. Huang, The Galerkin continuous finite element method for delay-differential equation with a variable term, Appl. Math. Comput., 186 (2007), 1488-1496.
doi: 10.1016/j.amc.2006.07.147. |
[12] |
Q. Huang, H. Xie and H. Brunner, Superconvergence of discontinuous Galerkin solutions for delay differential equation of pantograph type, SIAM. J. Sci. Comput., 33 (2011), 2664-2684.
doi: 10.1137/110824632. |
[13] |
A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38.
doi: 10.1017/S0956792500000966. |
[14] |
T. Koto, Stability of Runge-Kutta methods for the generalized pantograph equation, Numer. Math., 84 (1999), 233-247.
doi: 10.1007/s002110050470. |
[15] |
D. Li and C. Zhang, Superconvergence of a discontinuous Galerkin Method for first-order linear delay differential equations, J. Comput. Math., 29 (2011), 574-588.
doi: 10.4208/jcm.1107-m3433. |
[16] |
Y. Liu, On the $\theta$ -method for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190.
doi: 10.1016/0377-0427(95)00222-7. |
[17] |
N. Takama, Y. Muroya and E. Ishiwata, On the attainable order of collocation methods for delay differential equations with proportional delay, BIT Numer. Math., 40 (2000), 374-394.
doi: 10.1023/A:1022351309662. |
[18] |
X. Xu and Q. Huang, Continuous Galerkin methods for delay differential equations of pantograph type, Math. Pract. Theory, 24 (2014), 280-288 (in Chinese). |
[19] |
X. Xu, Q. Huang and H. Chen, Local superconvergence of continuous Galerkin solutions for delay differential equations of pantograph type, J. Comput. Math., 34 (2016), 186-199. |
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