Analytical and numerical dissipativity for nonlinear generalized pantograph equations

Previous Article
The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations
DCDS Home
This Issue
Next Article
Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets

July 2011, 29(3): 1245-1260. doi: 10.3934/dcds.2011.29.1245

Analytical and numerical dissipativity for nonlinear generalized pantograph equations

Wansheng Wang ^1, and Chengjian Zhang ^1,

1.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China, China

Received January 2010 Revised June 2010 Published November 2010

Abstract
Related Papers

This paper is concerned with the analytic and numerical dissipativity of nonlinear neutral differential equations with proportional delay, the so-called generalized pantograph equations. A sufficient condition for the dissipativity of the systems is given. It is shown that the backward Euler method inherits the dissipativity of the underlying system. Numerical examples are given to confirm the theoretical results.

Keywords: backward Euler method., Infinite delay, generalized pantograph equations, dissipativity.

Mathematics Subject Classification: Primary: 65L03, 65P40; Secondary: 38M0.

Citation: Wansheng Wang, Chengjian Zhang. Analytical and numerical dissipativity for nonlinear generalized pantograph equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1245-1260. doi: 10.3934/dcds.2011.29.1245

References:

[1]	Z. Cheng and C. M. Huang, Dissipativity for nonlinear neutral delay differential equations, J. Syst. Simul., 19 (2007), 3184-3187.
[2]	S. Q. Gan, Exact and discretized dissipativity of the pantograph equation, J. Comput. Math., 25 (2007), 81-88.
[3]	S. Q. Gan, Dissipativity of $\theta-$methods for nonlinear delay differential equations of neutral type, Appl. Numer. Math., 59 (2009), 1354-1365. doi: 10.1016/j.apnum.2008.08.003.
[4]	A. T. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.
[5]	A. T. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.
[6]	C. M. Huang, Dissipativity of Runge-Kutta methods for dynamical systems with delays, IMA J. Numer. Anal., 20 (2000), 153-166. doi: 10.1093/imanum/20.1.153.
[7]	C. M. Huang, Dissipativity of one-leg methods for dynamical systems with delays, Appl. Numer. Math., 35 (2000), 11-22. doi: 10.1016/S0168-9274(99)00048-3.
[8]	C. M. Huang, Dissipativity of multistep Runge-Kutta methods for dynamical systems with delays, Math. Comp. Model., 40 (2004), 1285-1296. doi: 10.1016/j.mcm.2005.01.019.
[9]	A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal., 31 (1994), 1452-1485. doi: 10.1137/0731075.
[10]	A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. doi: 10.1017/S0956792500000966.
[11]	H. Lehninger and Y. Liu, The functional-differential equation $y^'(t)=Ay(t)+By(\lambda t)+Cy^'(qt)+f(t)$, European J. Appl. Math., 9 (1998), 81-91. doi: 10.1017/S0956792597003343.
[12]	M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.
[13]	J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. A, 322 (1971), 447-468. doi: 10.1098/rspa.1971.0078.
[14]	A. M. Stuart and A. R. Humphries, Model problems in numerical stability theory for initial value problems, SIAM Review, 36 (1994), 226-257. doi: 10.1137/1036054.
[15]	A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis," Cambridge University Press, Cambridge, 1996.
[16]	H. J. Tian, Numerical and analytic dissipativity of the $\Theta-$method for delay differential equations with a bounde varible lag, International J. Bifur. Chaos, 14 (2004), 1839-1845. doi: 10.1142/S0218127404010096.
[17]	H. J. Tian, L. Q. Fan and J. X. Xiang, Numerical dissipativity of multistep methods for delay differential equations, Appl. Math. Comput., 188 (2007), 934-941. doi: 10.1016/j.amc.2006.10.048.
[18]	H. J. Tian and N. Guo, Asymptotic stability, contractivity and dissipativity of one-leg $\theta$-method for non-autonomous delay functional differential equations, Appl. Math. Comput., 203 (2008), 333-342. doi: 10.1016/j.amc.2008.04.045.
[19]	W. S. Wang and S. F. Li, On the one-leg $\theta$-methods for solving nonlinear neutral functional differential equations, Appl. Math. Comput., 193 (2007), 285-301. doi: 10.1016/j.amc.2007.03.064.
[20]	W. S. Wang and S. F. Li, Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay, Appl. Math. Lett., 21 (2008), 983-991. doi: 10.1016/j.aml.2007.10.014.
[21]	W. S. Wang and S. F. Li, Stability analysis of $\theta$-methods for nonlinear neutral functional differential equations, SIAM J. Sci. Comput., 30 (2008), 2181-2205. doi: 10.1137/060654116.
[22]	W. S. Wang, T. T. Qin and S. F. Li, Stability of one-leg $\theta$-methods for nonlinear neutral differential equations with proportional delay, Appl. Math. Comput., 213 (2009), 177-183. doi: 10.1016/j.amc.2009.03.010.
[23]	L. P. Wen and S. F. Li, Dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 324 (2006), 696-706. doi: 10.1016/j.jmaa.2005.12.031.
[24]	L. P. Wen, W. S. Wang and Y. X. Yu, Dissipativity of $\theta$-methods for a class of nonlinear neutral differential equations, Appl. Math. Comput., 202 (2008), 780-786. doi: 10.1016/j.amc.2008.03.022.
[25]	L. P. Wen, Y. X. Yu and W. S. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169-178. doi: 10.1016/j.jmaa.2008.05.007.
[26]	A. Xiao, Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces, Math. Numer. Sin. 22 (2000), 429-436.

show all references

References:

[1]	Z. Cheng and C. M. Huang, Dissipativity for nonlinear neutral delay differential equations, J. Syst. Simul., 19 (2007), 3184-3187.
[2]	S. Q. Gan, Exact and discretized dissipativity of the pantograph equation, J. Comput. Math., 25 (2007), 81-88.
[3]	S. Q. Gan, Dissipativity of $\theta-$methods for nonlinear delay differential equations of neutral type, Appl. Numer. Math., 59 (2009), 1354-1365. doi: 10.1016/j.apnum.2008.08.003.
[4]	A. T. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.
[5]	A. T. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.
[6]	C. M. Huang, Dissipativity of Runge-Kutta methods for dynamical systems with delays, IMA J. Numer. Anal., 20 (2000), 153-166. doi: 10.1093/imanum/20.1.153.
[7]	C. M. Huang, Dissipativity of one-leg methods for dynamical systems with delays, Appl. Numer. Math., 35 (2000), 11-22. doi: 10.1016/S0168-9274(99)00048-3.
[8]	C. M. Huang, Dissipativity of multistep Runge-Kutta methods for dynamical systems with delays, Math. Comp. Model., 40 (2004), 1285-1296. doi: 10.1016/j.mcm.2005.01.019.
[9]	A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal., 31 (1994), 1452-1485. doi: 10.1137/0731075.
[10]	A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. doi: 10.1017/S0956792500000966.
[11]	H. Lehninger and Y. Liu, The functional-differential equation $y^'(t)=Ay(t)+By(\lambda t)+Cy^'(qt)+f(t)$, European J. Appl. Math., 9 (1998), 81-91. doi: 10.1017/S0956792597003343.
[12]	M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.
[13]	J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. A, 322 (1971), 447-468. doi: 10.1098/rspa.1971.0078.
[14]	A. M. Stuart and A. R. Humphries, Model problems in numerical stability theory for initial value problems, SIAM Review, 36 (1994), 226-257. doi: 10.1137/1036054.
[15]	A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis," Cambridge University Press, Cambridge, 1996.
[16]	H. J. Tian, Numerical and analytic dissipativity of the $\Theta-$method for delay differential equations with a bounde varible lag, International J. Bifur. Chaos, 14 (2004), 1839-1845. doi: 10.1142/S0218127404010096.
[17]	H. J. Tian, L. Q. Fan and J. X. Xiang, Numerical dissipativity of multistep methods for delay differential equations, Appl. Math. Comput., 188 (2007), 934-941. doi: 10.1016/j.amc.2006.10.048.
[18]	H. J. Tian and N. Guo, Asymptotic stability, contractivity and dissipativity of one-leg $\theta$-method for non-autonomous delay functional differential equations, Appl. Math. Comput., 203 (2008), 333-342. doi: 10.1016/j.amc.2008.04.045.
[19]	W. S. Wang and S. F. Li, On the one-leg $\theta$-methods for solving nonlinear neutral functional differential equations, Appl. Math. Comput., 193 (2007), 285-301. doi: 10.1016/j.amc.2007.03.064.
[20]	W. S. Wang and S. F. Li, Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay, Appl. Math. Lett., 21 (2008), 983-991. doi: 10.1016/j.aml.2007.10.014.
[21]	W. S. Wang and S. F. Li, Stability analysis of $\theta$-methods for nonlinear neutral functional differential equations, SIAM J. Sci. Comput., 30 (2008), 2181-2205. doi: 10.1137/060654116.
[22]	W. S. Wang, T. T. Qin and S. F. Li, Stability of one-leg $\theta$-methods for nonlinear neutral differential equations with proportional delay, Appl. Math. Comput., 213 (2009), 177-183. doi: 10.1016/j.amc.2009.03.010.
[23]	L. P. Wen and S. F. Li, Dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 324 (2006), 696-706. doi: 10.1016/j.jmaa.2005.12.031.
[24]	L. P. Wen, W. S. Wang and Y. X. Yu, Dissipativity of $\theta$-methods for a class of nonlinear neutral differential equations, Appl. Math. Comput., 202 (2008), 780-786. doi: 10.1016/j.amc.2008.03.022.
[25]	L. P. Wen, Y. X. Yu and W. S. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169-178. doi: 10.1016/j.jmaa.2008.05.007.
[26]	A. Xiao, Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces, Math. Numer. Sin. 22 (2000), 429-436.

[1]	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
[2]	Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092
[3]	Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[4]	Qiumei Huang, Xiuxiu Xu, Hermann Brunner. Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5423-5443. doi: 10.3934/dcds.2016039
[5]	Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure and Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127
[6]	Jérémi Dardé, Sylvain Ervedoza. Backward uniqueness results for some parabolic equations in an infinite rod. Mathematical Control and Related Fields, 2019, 9 (4) : 673-696. doi: 10.3934/mcrf.2019046
[7]	Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379
[8]	Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585
[9]	Qi Lü, Xu Zhang. Transposition method for backward stochastic evolution equations revisited, and its application. Mathematical Control and Related Fields, 2015, 5 (3) : 529-555. doi: 10.3934/mcrf.2015.5.529
[10]	Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2733-2748. doi: 10.3934/cpaa.2014.13.2733
[11]	Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
[12]	Qi Lü, Xu Zhang. Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application. Mathematical Control and Related Fields, 2018, 8 (1) : 337-381. doi: 10.3934/mcrf.2018014
[13]	Yuncheng You. Pullback uniform dissipativity of stochastic reversible Schnackenberg equations. Conference Publications, 2015, 2015 (special) : 1134-1142. doi: 10.3934/proc.2015.1134
[14]	Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[15]	Jin Liang, James H. Liu, Ti-Jun Xiao. Condensing operators and periodic solutions of infinite delay impulsive evolution equations. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 475-485. doi: 10.3934/dcdss.2017023
[16]	Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272
[17]	Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[18]	Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[19]	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
[20]	Min Tang. Second order all speed method for the isentropic Euler equations. Kinetic and Related Models, 2012, 5 (1) : 155-184. doi: 10.3934/krm.2012.5.155

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Wansheng Wang Chengjian Zhang
on Google Scholar
Wansheng Wang Chengjian Zhang

[Back to Top]