中立型延迟积分微分方程数值方法的稳定性
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摘要
延迟积分微分方程广泛出现于物理、工程、生物、医学、航天航空及经济等领域,其算法理论研究具有毋庸置疑的重要性,近年来逐渐引起众多学者的极大关注.中立型延迟积分微分方程是一类重要的延迟积分微分方程.本文研究中立型延迟积分微分方程及数值方法的渐近稳定性.
近年来,有许多关于延迟积分微分方程数值方法的稳定性结果.然而,关于中立型延迟积分微分方程Runge-Kutta方法、多步Runge-Kutta方法及延迟积分微分方程Pouzet型线性多步方法的稳定性结论甚少.因此,研究这三类数值方法的稳定性具有重要意义.本文较系统地讨论这三类数值方法的稳定性.所获主要结果如下:
(1)讨论了Runge-Kutta方法求解中立型延迟积分微分方程的渐近稳定性,证明了A-稳定的Runge-Kutta方法能够保持原线性系统的渐近稳定性.
(2)讨论了多步Runge-Kutta方法求解中立型延迟积分微分方程的渐近稳定性,证明了A-稳定的多步Runge-Kutta方法能够保持原线性系统的渐近稳定性.
(3)讨论了Pouzet型线性多步方法求解延迟积分微分方程的渐近稳定性,证明了A-稳定且强零-稳定的Pouzet型线性多步方法能够保持原线性系统的渐近稳定性.
(4)通过数值例子和数值试验,对线性多步法及Runge-Kutta方法的稳定性进行了测试,测试结果进一步验证了本文所获理论结果的正确性.
Delay integro-differential equations (DIDEs) arise widely in the fields of Physics, Engineering, Biology, Medical Science, Aviation, Economics and so on. The theory of computational methods is very important in DIDEs. Recently, many scholars have pay careful attention to it. Neutral delay integro-differential equations (NDIDEs) is special subclass of DIDEs. In this paper, we research the asymptotic stability of NDIDEs and numerical methods for NDIDEs.
Recently, there are many results of the stability of numerical methods for DIDEs. But, there are few results of the stability of the Runge-Kutta methods and the multistep Runge-Kutta methods of NDIDE s and the linear multistep methods of Pouzet type for DIDEs. Therefore, it is significant to study the stability of these three classes of numerical methods. In this article, we systematically study the stability of these three classes of numerical methods. The main results in the thesis are as follows:
(1) We investigate the asymptotic stability of the Runge-Kutta methods of NDIDEs. It is proven that A-stable Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems.
(2) We study the asymptotic stability of the multistep Runge-Kutta methods of NDIDEs. It is shown that A-stable multistep Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems.
(3) We analyze the asymptotic stability of the linear multistep methods of Pouzet type for DIDEs. It is proven that A-stable and strongly zero-stable linear multistep methods of Pouzet type can preserve the asymptotic stability of the underlying linear systems.
(4) Numerical examples and numerical experiments are given for checking the stability of the linear multistep methods and the Runge-Kutt a methods, which confirm the theoretical results obtained in this paper.
近年来,有许多关于延迟积分微分方程数值方法的稳定性结果.然而,关于中立型延迟积分微分方程Runge-Kutta方法、多步Runge-Kutta方法及延迟积分微分方程Pouzet型线性多步方法的稳定性结论甚少.因此,研究这三类数值方法的稳定性具有重要意义.本文较系统地讨论这三类数值方法的稳定性.所获主要结果如下:
(1)讨论了Runge-Kutta方法求解中立型延迟积分微分方程的渐近稳定性,证明了A-稳定的Runge-Kutta方法能够保持原线性系统的渐近稳定性.
(2)讨论了多步Runge-Kutta方法求解中立型延迟积分微分方程的渐近稳定性,证明了A-稳定的多步Runge-Kutta方法能够保持原线性系统的渐近稳定性.
(3)讨论了Pouzet型线性多步方法求解延迟积分微分方程的渐近稳定性,证明了A-稳定且强零-稳定的Pouzet型线性多步方法能够保持原线性系统的渐近稳定性.
(4)通过数值例子和数值试验,对线性多步法及Runge-Kutta方法的稳定性进行了测试,测试结果进一步验证了本文所获理论结果的正确性.
Delay integro-differential equations (DIDEs) arise widely in the fields of Physics, Engineering, Biology, Medical Science, Aviation, Economics and so on. The theory of computational methods is very important in DIDEs. Recently, many scholars have pay careful attention to it. Neutral delay integro-differential equations (NDIDEs) is special subclass of DIDEs. In this paper, we research the asymptotic stability of NDIDEs and numerical methods for NDIDEs.
Recently, there are many results of the stability of numerical methods for DIDEs. But, there are few results of the stability of the Runge-Kutta methods and the multistep Runge-Kutta methods of NDIDE s and the linear multistep methods of Pouzet type for DIDEs. Therefore, it is significant to study the stability of these three classes of numerical methods. In this article, we systematically study the stability of these three classes of numerical methods. The main results in the thesis are as follows:
(1) We investigate the asymptotic stability of the Runge-Kutta methods of NDIDEs. It is proven that A-stable Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems.
(2) We study the asymptotic stability of the multistep Runge-Kutta methods of NDIDEs. It is shown that A-stable multistep Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems.
(3) We analyze the asymptotic stability of the linear multistep methods of Pouzet type for DIDEs. It is proven that A-stable and strongly zero-stable linear multistep methods of Pouzet type can preserve the asymptotic stability of the underlying linear systems.
(4) Numerical examples and numerical experiments are given for checking the stability of the linear multistep methods and the Runge-Kutt a methods, which confirm the theoretical results obtained in this paper.
引文
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[2]秦元勋,刘永清,王联,郑祖麻.带有时滞的动力系统的运动稳定性.北京:科学出版社,1989.69-83
[3]刘永清.滞后大系统的工程实际与稳定性方法.北京:科学出版社,1992.23-27
[4]J.D.Murray.Mathematical Biology.Berlin:Springer-Verlag,1989.46-49
[5]R.D.Driver.Ordinary and Delay Differential Equations.Berlin:Springer-Verlag,1977.1-117
[6]J.K.Hale.Theory of Functional Differential Equations.New York:Springer-Verlag,1977.1-179
[7]Y.Kuang.Delay Differential Equations with Applications in Population Dynamics.Boston:Academic Press,1993.1-395
[8]G.I.Marchuk.Mathematical Modelling of Immune Response in Infec tions Diseases.Dordrecht:Kluwer,1997.1-77
[9]C.T.H.Baker,G.A.Bocharov and F.A.Rihan.A Report on the Use of Delay Differential Equations in Numerical Modelling in the Biosciences.Manchester:MCCM Technical Report,1999,(343):2-6
[10]J.T.Ockendon,A.B.Tayer.The Dynamics of a Current Collection System for an Electric Locomotive.Proc.Roy.Soc.London Ser.1971,(322):447-468
[11]L.Kenneth,K.L.Cooke and W.Joseph.Retarded Differential Equa tions with Piecewise Constant Delays.J.Math.Anal.Appl.1984,(99):265-297
[12]M.Zennaro.Delay Differential Equations:Theory and Numerics.Oxford:Oxford University Press,1995.291-333
[13]W.P.London,J.A.Yorke.Recurrent Epidemics of Measles,Chicken pox and Mumps I:Seasonal Variation in Contact Rates.Amer.J.Epid.1973,(98):469-482
[14]E.A.Grove,G.Ladas and C.Qian.Global Attractivity in a "Food- limited" Population Model.Dymanic System Appl.1993,(2):243-250
[15]F.L.Willian.Differential Equation Models.New York:Springer-Verlag,1983.37-46
[16]钱学森.工程控制论.北京:科学出版社,1958.1-78
[17]陆启韶.常微分方程的定性方法和分叉.北京:北京航空航天大学出版社,1989.47-59
[18]Y.S.Chin.On the Equivalence Problem of Differential Equations and Difference-differential Equation in the Thoery of Stability.Sci.Record.1957,(1):287-289
[19]Y.S.Chin,L.G.Liou and L.Wang.Effect of Time-lags on Stability of Dynamical System.Sci.Sinica.1960,(9):719-747
[20]王联.稳定性理论中第一临界情形的微分方程与微分差分方程的等价性问题.数学学报.1960,(10):104-124
[21]R.Bellman,K.L.Gooke.Differential-difference Equations.New York:Academic Press,1963.1-461
[22]R.K.Brayton,R.A.Willoughby.On the Numerical Integration of a Symmetric System of Difference-differential Equations of Neutral Type.J.Math.Anal.Appl.1967,(18):182-189
[23]V.K.Barwell.Special Stability Problems for Functional Differe ntial Equations.BIT.1975,(15):130-135
[24]J.Mori,E.Noldus and M.Kuwahara.A Way to Stabilize Linear System with Delayed State.Automatica.1983,(19):571-573
[25]A.Bellen,Z.Jackiewicz and M.Zennaro.Stability Analysis of One-step Methods for Neutral Delay-differential Equations.Numer.Math.1988,(52):605-619
[26]K.J.in' t Hour.The Stability of θ-methods for Systems of Delay Differential Equations.Ann.Numer.Math.1994,(1):323-334
[27]J.X.Kuang,J.X.Xiang and H.J.Tian.The Asymptotic Stability of One-parameter Methods for Neutral Differential Equations.BIT.1994,(34):400-408
[28]G.D.Hu,B.Cahlon.Estimations on Numerically Stable Step-size for Neutral Delay Differential Systems with Multiple Delays.J.Comput.Appl.Math.1999,(102):221-234
[29]T.Kato,J.B.Mcleod.The Functional-differential Equation y'(x) = ay(λx) +by(x). Bull. Amer. Math. Soc. 1971, (77) :891-937
[30] LTorelli. Stability of Numerical Methods for Delay Differential Equations. J. Comput. App. Math. 1989, (25):15-26
[31] A.Iserles. On Nonlinear Delay Differential Equations. Trans.Amer. Math. Soc. 1994, (1):441-477
[32] Y.K.Liu. Stability Analysis of θ-methods for Neutral Functional-differential Equations. Numer.Math. 1995, (70):473—485
[33] L. Qiu, T.Mitsui and J. X.Kuang. The Numerical Stability of the θ-methods for Delay Differential Equations with Many Variable Delays.J. Comput. Math. 1999, (17):523-532
[34] T. Koto. Stability of Runge-Kutta methods for delay integro-differential equations. J.Comput.Appl.Math.2002, (145):483—492
[35] C.M.Huang, S. Vandewalle. An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIMA:J. Sci. Comput, 2003.1—233
[36] J.J.Zhao, Y. Xu and M.Z.Liu. Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system.Appl. Math. Comput. 2005, (167):1062-1079
[37] E. Hairer, S. P. Norsett and G. Wanner. Solving Ordinary Differential Equations I, 2nd, Berlin:Springer-Verlag, 1996.1—312
[38] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II,2nd Berlin:Springer-Verlag, 1996. 1—221
[39] J. C. Butcher. Numerical Methods for Ordinary Differential Equations in the 20th Century. J. Comput. Appl. Math. 2000, (125): 1—29
[40] J.D.Lambert. Computational methods in ordinary differential equations. Scotland:Dundee, 1990.1—236
[41] R. Bellman. On the Computational Solution of Differential-difference Equations. J. Math. Anal. Appl. 1961, (2):108—110
[42] R. Bellman, K. L. Cooke. On the Computational Solution of a Class of Functional Differential Equations. J. Math. Anal. Appl. 1965, (12) :495—500
[43] C. W.Cryer. A New Class of Highly Stable Methods 4-stable Methods. BIT. 1973, (13):153—159
[44] M.Z.Liu, M. N. Spijker. The Stability of theθ-methods in the Numerical Solution of Delay Differential Equations.IMA J.Numer.Anal.1990,(10):31-48
[45]T.Kato.A Stability Property of A-stable Natural Runge-Kutta Methods for Systems of Delay Differential Equations.BIT.1994,(34):262-267
[46]K.J.in' t Hout,Stability Analysis of Runge-Kutta Methods for Systems of Delay Differential Equations,IMA J.Numer.Anal.1997,(17):17-27
[47]张诚坚,周叔子.中立型多滞量微分方程系统的理论解与数值解的渐近稳定性.中国科学.1998,(28):713-720
[48]P.J.van der Houwen,B.P.Smmeijer.Stability in Linear Multistep Methods for Pure Delay Equations.J.Comput.Appl.Math.1984,(10):55-63
[49]L.rorelli.A Sufficient Condition for GPN-stability for Delay Differential Equations.Numer.Math.1991,(59):311-320
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