奇异摄动延迟积分微分方程的线性多步法
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摘要
奇异摄动问题广泛地存在于化学动力学、自动控制、电子系统等一系列高科技领域的数学模型中,这类问题往往具有重要性和特殊的计算复杂性,发展和深化针对各种数值方法的定量收敛理论受到了各国科学家的广泛关注。
奇异摄动初值问题是一类特殊的刚性问题,其特殊的结构特点使得这类问题不能被B-理论满意地覆盖。这一问题主要包括单刚性(刚性仅由小参数引起)和多刚性(刚性由小参数和其它因素引起)奇异摄动初值问题两大类。目前,对于最简单形式的单参数奇异摄动初值问题的数值方法国内外已有一些研究,Hairer,Lubich,Strehmel,Schneider,肖爱国,甘四清和孙耿等人基于直接法或ε-渐进展开法两种不同的方法分别给出了许多关于线性多步方法,Runge-Kutta方法,Rosenbrock型方法,一般线性方法等关于奇异摄动经典初值问题的收敛性结果。然而对于同时具备离散型延迟和分布型延迟的一类奇异摄动积分微分系统初值问题数值方法收敛性的研究,国内外尚未见到这方面的结果。因此,对该领域的研究无疑具有重要的理论意义和广阔的应用前景。
本文首先探讨了一类延迟奇异摄动积分微分系统的指数稳定性质,第三章和第四章分别致力于讨论线性多步方法关于单刚性和多刚性两类问题的定量收敛性,给出了其整体误差分析。每章结尾的数值试验结果证明了本文理论结果的正确性。
Singular perturbation problems widely arise in many practical applications, such aschemical kinetics, automatic control, and electrical circuits et.al. By view of the specialcomplexity and importance of this problems, many scientists from all over the worldhave paid extensive attention to develop and deepen quantitative convergence theory ofnumerical methods for these important typical subclasses of stiff problems.
Initial value problems of ordinary differential equations in singular perturbation formare of the class of stiff problems which can’t be satisfactorily covered by B-theory becauseof their very special structure. They mainly include two subclasses: singly stiff singularperturbation problems (their stiffness is only caused by some small parameters) and multi-ply stiff singular perturbation problems (their stiffness is caused by some small parametersand some other factors). At present, there exist many researches into numerical methodsfor typical singular perturbation problems. Many important and interesting results on theconvergence of linear multistep methods, RungeffKutta methods, Rosenbrock methodsand general linear methods applied to the classical singular perturbation problems havebeen given by many authors by means of two different approaches ff direct approach and-expansion approach.
But up to now, there exits no results of numerical methods for singular perturbationintegral differential problems with delays which often arise in many practical applications,such as chemical kinetics, automatic control, electrical circuits and nuclear reactor kineticset.al. Therefore, a deep research into this field is of important theoretical meanings andvast applied prospects.
In this paper, we first proved the stability property of a class of singular perturba- tion delay volterra integral differential systems(SPDVIDs). Then, we have gained somequantitative convergence results for linear multistep methods applied to both singly stiffsingular perturbation problems and multiply stiff singular perturbation problems in chap-ter 3 and 4. In the end of each chapter, we also gave some numerical test which confirmedour theoretical results.
奇异摄动初值问题是一类特殊的刚性问题,其特殊的结构特点使得这类问题不能被B-理论满意地覆盖。这一问题主要包括单刚性(刚性仅由小参数引起)和多刚性(刚性由小参数和其它因素引起)奇异摄动初值问题两大类。目前,对于最简单形式的单参数奇异摄动初值问题的数值方法国内外已有一些研究,Hairer,Lubich,Strehmel,Schneider,肖爱国,甘四清和孙耿等人基于直接法或ε-渐进展开法两种不同的方法分别给出了许多关于线性多步方法,Runge-Kutta方法,Rosenbrock型方法,一般线性方法等关于奇异摄动经典初值问题的收敛性结果。然而对于同时具备离散型延迟和分布型延迟的一类奇异摄动积分微分系统初值问题数值方法收敛性的研究,国内外尚未见到这方面的结果。因此,对该领域的研究无疑具有重要的理论意义和广阔的应用前景。
本文首先探讨了一类延迟奇异摄动积分微分系统的指数稳定性质,第三章和第四章分别致力于讨论线性多步方法关于单刚性和多刚性两类问题的定量收敛性,给出了其整体误差分析。每章结尾的数值试验结果证明了本文理论结果的正确性。
Singular perturbation problems widely arise in many practical applications, such aschemical kinetics, automatic control, and electrical circuits et.al. By view of the specialcomplexity and importance of this problems, many scientists from all over the worldhave paid extensive attention to develop and deepen quantitative convergence theory ofnumerical methods for these important typical subclasses of stiff problems.
Initial value problems of ordinary differential equations in singular perturbation formare of the class of stiff problems which can’t be satisfactorily covered by B-theory becauseof their very special structure. They mainly include two subclasses: singly stiff singularperturbation problems (their stiffness is only caused by some small parameters) and multi-ply stiff singular perturbation problems (their stiffness is caused by some small parametersand some other factors). At present, there exist many researches into numerical methodsfor typical singular perturbation problems. Many important and interesting results on theconvergence of linear multistep methods, RungeffKutta methods, Rosenbrock methodsand general linear methods applied to the classical singular perturbation problems havebeen given by many authors by means of two different approaches ff direct approach and-expansion approach.
But up to now, there exits no results of numerical methods for singular perturbationintegral differential problems with delays which often arise in many practical applications,such as chemical kinetics, automatic control, electrical circuits and nuclear reactor kineticset.al. Therefore, a deep research into this field is of important theoretical meanings andvast applied prospects.
In this paper, we first proved the stability property of a class of singular perturba- tion delay volterra integral differential systems(SPDVIDs). Then, we have gained somequantitative convergence results for linear multistep methods applied to both singly stiffsingular perturbation problems and multiply stiff singular perturbation problems in chap-ter 3 and 4. In the end of each chapter, we also gave some numerical test which confirmedour theoretical results.
引文
[1] C. T. H. Baker, N. J. Ford.Convergence of linear multistep methods for a class ofdelay integro-di?erential equations. in R. P. Agarwal, Y. M. Chow, S. J. Wilson(Eds),Inter Serie Numer. Math. 1988, 86: 47-59
[2] C. T. H. Baker et al.A Report on the Use of Delay Di?erential Equations inNumerical Modelling in the Biosciences. MCCM Technical Report 343 ISSN. 1999,1360-1725
[3] L. Blank.Stability result for collocation methods for Volterra integral equations.Appl. Math. Comp. 1996, 79: 267-288
[4] L. Blank.Stability of collocation for weakly singular Volterra equations. IMA. J.Numer. Anal. 1995, 15: 357-375
[5] R. Bellman, K. L. Cooke.Di?erential-Di?erence Equations. Academic Press.New York San Francisco London. 1963, 440-455
[6] Bellen A., Zennaro M..Numerical solution of delay di?erential equations by uni-form corrections to an implicit Runge-Kutta method. Numer. Math, 1985, 47:301-316
[7] A. Bellen, Z. Jackiewicz, M. Zennaro.Stability Analysis of One-Step Methodsfor Neutral Delay Di?erential Equations. Numer. Math. 1988, 52: 605-619
[8] A. Bellen, N. Guglielmi, M. Zennaro.On the Contractivity and Asymptotic-Stability of Systems of Delay Di?erential Equations of Neutral Type. BIT. 1999, 39:1-24
[9] A. Bellen, M. Zennaro.Strong Contractivity Properties of Numerical Methods forOrdinary and Delay Di?erential Equations. App. Numer. Math. 1992, 9: 321-346
[10] V. K. Barwell.On the Asymptotic Behavior of the Solution of Di?erential Di?er-ence Equations. Utilitas. Math. 1974, 6: 189-194
[11] H. Brunner.The numerical solutions of neutral Volterra integro-di?erential equa-tions with delay argument. Appl. Numer. Math. 1994, 1:309-322
[12] B. Cahlon.On the numerical stability of Volterra integral equations with delay ar-guments. J. Comput. Appl. Math. 1990, 33:97-104
[13] B. Cahlon.On the stability of Volterra integral equations with a lagging argument.BIT. 1995, 35:19-29
[14] C. Lubich.On the convergence of multistep methods for nonlinear sti? di?erentialequations. Numer. Math., 58(1991)839-853
[15] M.R. Crisci, E. Russo.A Vecchio stability results for one step discretized colloca-tion methods in the numerical treatment of Volterra integral equations. Math. Comp.1992, 58:119-134
[16] E. Hairer, G. Wanner.Solving ordinary di?erential equations Π Sti? anddi?erential-algebraic problems. Spinger-Verlag, Berlin, 1991
[17] E. Hairer.A- and B-stability for Runge-Kutta methods characterizations and equiv-alence. Numer. Math, 48(1986), 383-389
[18] H. Brunner, E. Hairer, S. P. Nprsett.Runge-Kutta theory for Volterra integralequations of the second kind. Math. Comp, 1982, 39: 147-163
[19] W. H. Enright, Min. Hu.Continuous Runge-Kutta methods for neutral Volterraintegro-di?erential equations with delay. Apll. Numer. Math. 1997, 24:175-190
[20] Enright W. H., Hayashi H..Convergence analysis of the solution of retarted andneutral delay di?erential equations by continuous numerical methods. SIAM. J. Nu-mer. Amal. 1998, 35:572-585
[21] N. J. Ford, C. T. H. Baker.Qualitative behaviour and stability of solutions ofdiscretised nonlinear Volterra integral equations of convolution type. J. Comp. Appl.Math. 1996, 66:213-225
[22] G. Wanner, E. Hairer, S. P. Norsett.Order star and stability theorems. BIT.18(1978): 475-489
[23] G. Dahlquist.Error analysis for a class of methods for sti? nonlinear initial valueproblems. Numer. Anal. Dundee, Lecture Notes in Math 506, Berlin: Springer-Verlag,1975, 60-74
[24] G. Dahlquist.G-stability is equivalent to A-stability. BIT, 18(1978): 384-401
[25] J. K. Hale.Theory of Functional Di?erential Equations. Springer, New York, 1977
[26] J. K. Hale.Homoclinic Orbits and Chaos in Delay Equations. In proceedings of theNinth Dundee Conference on Ordinary and Partial Di?erential Equations, edited byB. D. Sleeman et al, New York, 1986
[27] Huang Chengming.B-convergence properties of algebraically stable general linearmethods for strictly dissipative initial value problems. Proc. Int. Conf. on Comput.Di?erential Equations and Dynamical Systems, Beijing, 8-79, 1992
[28] Huang C. M., Li. S. F., Fu. H. Y..Stability and error analysis of one-leg methodsfor nonlinear delay di?erential equations. J. Comput. Appl. Math, 1999, 103:263-279
[29] C. M. Huang, S. Li, H. Fu, G. Chen.Stability Analysis of Runge-Kutta Methodsfor Nonlinear Delay Di?erential Equations. BIT. 1999, 39: 270-280
[30] C. M. Huang, S. Li, H. Fu, G. Chen.Stability and Error Analysis of One-LegMethods for Nonlinear Delay Di?erential Equations. J. Comput Appl. Math. 1999,103: 263-279
[31] C. M. Huang, Q. Chang.Linear Stability of General Linear Methods for Systemsof Neutral Delay Di?erential Equations. Appl. Math. Letters. 2001, 14: 1017-1021
[32] C. M. Huang, V. Stefan. Stability of Runge-Kutta-Pouzet methods for Volterraintegral and integro-di?erential equations with delays. Rep. TW361, K. U. Leuven.2003, 6
[33] C. Huang, S.Vandewalle. An Analysis of Delay-Depentant Stability for Ordinaryand Patial Di?erential Equations with Fixed and Distributed Delays. SIAM J. SciComput. 2004, 25(5): 1608-1632
[34] K. J. int’Hout, M. N. Spijker.The θ-methods in the Numerical Solution of DelayDi?erential Equations. Proc. Interat. Sem. NUMDIFF-5. Leipzig. 1989
[35] K. J. int’Hout, M. N. Spijker.Stability Analysis of Numerical Methods for DelayDi?erential Equations. Numer. Math. 1991, 59: 804-814
[36] K. J. int’Hout.The Stability of A Class of Runge-Kutta Methods for Delay Di?er-ential Equations. Appl. Numer. Math. 1992, 9: 347-355
[37] K. J. int’Hout, M. N. Spijker.A New Interpolation Procedure for AdaptingRunge-Kutta Methods for Delay Di?erential Equations. BIT. 1992, 32: 643-649
[38] K. J. int’Hout.The Stability of the θ-methods for Systems of Delay Di?erentialEquations. Report Series. Univ. Auckland. 1993, No.282
[39] K. J. int’Hout.A Note on Unconditional Maximum Norn Contrativity of Diago-nally Split Runge-Kutta Methods. SIAM. J. Numer. Anal. 1996, 33: 1125-1134
[40] K. J. int’Hout, M. N. Spijker.Stability Analysis of Runge-Kutta Methods forSystems of Delay Di?erential Equations. IMA. J. Numer. Anal. 1997, 17: 17-27
[41] G. D. Hu, T. Mitsui.Stability Analysis of Numerical Methods for Systems of Neu-tral Delay-Di?erential Equations. BIT. 1995, 35: 504-515
[42] G. D. Hu, B. Cahlon.Numerically Stable Step-Size for Neutral Delay Di?erentialSystems with Multiple Delays. J. Comput Appl. Math. 1999, 102: 221-234
[43] Z. Jackiewicz.Asymptotic Stability Analysis of θ-Methods for Functional Di?eren-tial Equations. Numer. Math. 1984, 43: 389-396
[44] J. C. Butcher.A stability property of implicit Runge-Kutta methods. BIT, 15(1975):358-361
[45] J. C. Butcher.Linear and nonlinear stability for general linear methods. BIT.27(1987): 182-189
[46] J. D. Lambert.Computational methods in ordinary di?erential equations. John Wi-ley and Sons Ltd, 1973
[47] L. Kenneth, K. L. Cooker, W. Joseph.Retarded Di?erential Equations withPiecewise Constant Delays. J. Math. Anal. and Appl. 1984, 99: 265-297
[48] K. Burrage, J. C. Butcher.Stability criteria for implicit Runge-Kutta methods.SIAM. J. Numer. Anal. 16(1979): 46-57
[49] K. Dekker, J. G. Verwer.Stability of Runge-Kutta Methods for Sti? NonlinearDi?erential Equations. North-Holland, Amsterdam, 1984
[50] K. Strehmel, R. Weiner, I. Dannehl.On error behavior of partitioned lin-early implicit Runge-Kutta methods for sti? and di?erential algebraic systems. BIT,30(1990), 358-375
[51] T. Koto.A Stability Property of A-stable natural Runge-Kutta methods for Systemsof Delay Di?erential Equations. BIT. 1994, 34: 262-267
[52] T. Koto.Stability of Runge-Kutta methods for delay integro-di?erential equations.J. Comp. Appl. Math. 2002, 45: 483-495
[53] J. X. Kuang, H. J. Tian.The Asymptotic Behaviour of Theoretical and Numericalfor the De?erential Equations with Several Delay Terms. J. Shanghai Normal Univ.1994, 23: 1-10
[54] J. X. Kuang, H. J. Tian.The Asymptotic Analysis of One-Parameter Methods forNeutral Delay Di?erential Equations. BIT. 1994, 34: 400-408
[55] L. F. Shampine, Gear C. W.A user’s view of solving sti? ordinary di?erentialequations. SIAM Review. 1979, 21:1-17
[56] Li Shoufu.B-convergence of general linear methods. Proc. Bail-V. Int.Conf(Shanghai, 1988), 203-208
[57] Li Shoufu.Stability and B-convergence of general linear methods. J. Comput. Appl.Math, 28(1989), 565-575
[58] Li Shoufu.Nonlinear stability of general linear methods. J. Comput. Math,9(2)(1991), 97-104
[59] Li Shoufu.B-convergence of general multivalue methods applied to sti? problems inBanach spaces. Science In China(Series A), 36(1993), 1-13
[60] Li Shoufu.B-convergence properties of multistep Runge-Kutta methods. Math. Com-put. 62(1994), 565-575
[61] Li Shoufu.Theory of computational methods for sti? di?erential equations. HunanSci. and Tech. Press, 1997(in Chinese)
[62] M. Z. Liu, M. N. Spiijker.The Stability of the θ-methods in the Numerical Solutionof Delay Di?erential Equations. IMA. J. Numer. Anal. 1990, 10: 31-48
[63] J. Mori, E. Noldus, M. Kuwahara.A Way to Stabilize Linear Systems withDelayes State. Automatica. 1983, 19:571-573
[64] Y. S. Chin.On the Equivalence Problem of Di?erential Equations and Di?erenceEquations in the Theory of Stability. Sci. Record. 1957, 1:287-289
[65] Y. S. Chin et al.E?ect of Time-lags on Stability of Dynamical Systems. Sci. Sinica.1960, 9:719-747
[66] L. Qiu et al.The NGP-Stability of Runge-Kutta Methods for Systems of NeutralDelay Di?erential Equations. Numer. Math. 1999, 81: 451-459
[67] R. Frank, J. Schneid, C. W. Ueberhuber.The concept of B-convergence. SIAM.J. Numer. Anal, 18(1981), 753-780
[68] R. Frank, J. Schneid, C. W. Ueberhuber.Order results for implicit Runge-Kutta methods applied to sti? systems. SIAM. J. Numer. Anal, 22(1985), 515-534
[69] R. Frank, J. Schneid, C. W. Ueberhuber.Stability properties of implicit Runge-Kutta methods. SIAM. J. Numer. Anal, 22(1985), 497-515
[70] R. Frank, J. Schneid, H. Lehner.B-convergence properties of defect correctionmethods. |,Π. Numer. Math, 46(1986), 139-188
[71] S. Schneider.Convergence results for general linear methods on singular perturba-tion problems. BIT. 33(1993), 670-686
[72] Siqing Gan, Geng Sun, Weimin Zheng.Errors of Linear Methods and Runge-Kutta Methods for Singular Perturbation Problems with Delays. J. Comput. Appl.Math., 44(2002)1157-1173
[73] H. J. Tian, J. X. Kuang.The Stability of Linear Multistep Methods for Systemsof Delay Di?erential Equations. Numer. Math. A. Journal of Chinese Univ. 1995, 4:10-16
[74] H. J. Tian, J. X. Kuang.The numerical stability analysis of Runge-Kutta Methodsfor Volterra integral equations with delay terms. J. AMM. 1995, 16: 485-491
[75] H. J. Tian, J. X. Kuang.The Stability Analysis of the θ-methods for Delay Dif-ferential Equations. J. Comput. Math. 1996, 14: 203-212
[76] H. J. Tian, J. X. Kuang.The Numerical Stability of Linear Multistep Methodsfor Delay Di?erential Equations with Several Delays. SIAM. J. Numer. Anal. 1996,33(3): 883-889
[77] H. Tian.Numerical treatment of singularly perturbed delay di?erential equations.Ph.D. thesis, University of Manchester(2000).
[78] Hongjiong Tian.The exponential asymptotic stability of singularly perturbed delaydi?erential equations with a bounded lag. J. Math. Anal. Appl. 270(2002)143-149
[79] C.T.H. Baker, C.A.H. Paul, H. Tian.Di?erential algebraic equations with after-e?ect. J. Comput. Appl. Math. 140(2002):63-80
[80] L. Torelli.Stability of Numerical Methods for Delay De?erential Equations. J.Comput. Appl. Math. 1989, 25: 15-26
[81] L. Torelli. Stability of Numerical Methods for Delay Di?erential Equations Acti-vated Tanks. J. Comput. Appl. Math. 1989, 25: 15-26
[82] R. Verminglio, L. Torelli. A Stable Numerical Approach for Implicit Non-Linearneutral Delay Di?erential Equations. BIT. 2003, 43: 195-215
[83] R. Vermiglio.On the stability of Runge-Kutta methods for delay integral equations.Numer. Math. 1992, 61: 561-577
[84] J. Wiener.Retarded Di?erential Equations with Piecewise Constant Delays. J. Math.Anal. Appl. 1984, 99: 265-297
[85] D. S. Watanable, M. G. Roth.The Stability of Di?erence Formulas for DelayDi?erential Equations. SIAM. J. Numer. Anal. 1985, 22: 132-145
[86] W. Auzinger, R. Frank, G. Kirlinger.Extending convergence theory for non-linear sti? problems. Part1. BIT. 36:4, 1996, 635-652
[87] Xiao Aiguo.An algebraic characterization of B-convergent general Linear methodsfor strictly dissipative initial value problems. Proc. 3rd. Int. Confon. Syst. Simul. Sci.Comput, Bejing Int. Acad. Pub. 135-137, 1995
[88] Xiao Aiguo, Li Shoufu, Fu hongyuan, Chen Guangnan.Convergence ofLinear multistep methods for two-parameter singular perturbation problems. ACTAMATHEMATICAE APPLICATAE SINICA, Vol 17, No.2, 2001
[89] B. Yang, L. Qiu, T. Mitsui.GPk-Stability of Runge-Kutta Methods for GeneralizedDelay Di?erential Systems. Comput. Math. Appl. 1999, 37: 89-97
[90] M. Zennaro.P-Stability Properties of Runge-Kutta Methods for delay Di?erentialEquations. Numer. Math. 1986, 49: 302-318
[91] M. Zennaro.Contracticity of Runge-Kutta Methods with Respect to Forcing Terms.Appl. Numer. Math. 1993, 10: 321-345
[92] Zhang C. J., Zhou. S. Z.Nonlinear stability and D-convergence of Runge-Kuttamethods for delay di?erential equations. J. Comput. Appl. Math. 1997, 85:225-237
[93] C. J. Zhang, X. X. Liao, W. Cheng.GPk-stability of the Multistep Runge-KuttaMethods for a Class of Multidelay Systems. Math. Appl. 2000, 13(3): 139-142
[94] C. J. Zhang, X. X. Liao. Stability of BDF methods for Volterra integral equationswith delay. Comp. Math. Appl. 2002, 43:95-102
[95] C. Zhang, S. Vandewalle. Stability Analysis of Volterra Delay-Integro-Di?erential Equations and Their Backward Di?erentiation Time Discretization. J.Comput. App. Math. 2004, 164/165: 797-814
[96] C. J. Zhang, V. Stefan. Stability analysis of Runge-Kutta methods for nonlinearVolterra delay di?erential equations. IMA. J. Numer. Anal. 2004, 24: 193-214
[97] 李寿佛. Banach空间中非线性刚性Volterra泛函微分方程稳定性分析. 中国科学.2005, 35(3), 286-301
[98] 侯文星.一类奇异摄动初值问题BDF方法的收敛性. 湘南学院学报,2004. Vol.25No.5:35-36
[99] 徐阳,赵景军,刘明珠. 二阶延迟微分方程θ -方法的TH-稳定性. 计算数学. 2004,26(2), 189-192
[100] 王晚生,李寿佛. 非线性中立型延迟微分方程稳定性分析. 计算数学. 2004, 26(3),303-314
[101] 郭长勇. 几类延迟微分方程的数值稳定性. 哈尔滨工业大学硕士论文. 2002
[102] 甘四清,孙耿.时滞奇异摄动问题单支方法的收敛性. 中国科学(A辑). 2001,31(4): 314-323
[103] 张诚坚, 高健.隐式Euler方法关于V olterra延迟积分方程的数值稳定性. 应用数学,2000, 13(4):130-132
[104] 范振成.二阶延迟微分方程数值稳定性分析. 哈尔滨工业大学硕士论文. 2000
[105] W. Cucas.微分方程模型. 国防科技大学出版社. 1999
[106] 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(A辑). 1998, 28(8): 713-720
[107] 李寿佛.刚性微分方程算法理论. 长沙:湖南科学技术出版社,1997
[108] 刘德贵,费景高,韩天敏.刚性大系统数字仿真方法. 河南科学技术出版社,1996
[109] 秦元勋等.带有时滞的动力系统的运动稳定性. 科学出版社. 1992
[110] 刘永清.滞后大系统的工程实际与稳定性方法. 科学出版社. 1992
[111] 钱学森.工程控制论. 科学出版社. 1958
[2] C. T. H. Baker et al.A Report on the Use of Delay Di?erential Equations inNumerical Modelling in the Biosciences. MCCM Technical Report 343 ISSN. 1999,1360-1725
[3] L. Blank.Stability result for collocation methods for Volterra integral equations.Appl. Math. Comp. 1996, 79: 267-288
[4] L. Blank.Stability of collocation for weakly singular Volterra equations. IMA. J.Numer. Anal. 1995, 15: 357-375
[5] R. Bellman, K. L. Cooke.Di?erential-Di?erence Equations. Academic Press.New York San Francisco London. 1963, 440-455
[6] Bellen A., Zennaro M..Numerical solution of delay di?erential equations by uni-form corrections to an implicit Runge-Kutta method. Numer. Math, 1985, 47:301-316
[7] A. Bellen, Z. Jackiewicz, M. Zennaro.Stability Analysis of One-Step Methodsfor Neutral Delay Di?erential Equations. Numer. Math. 1988, 52: 605-619
[8] A. Bellen, N. Guglielmi, M. Zennaro.On the Contractivity and Asymptotic-Stability of Systems of Delay Di?erential Equations of Neutral Type. BIT. 1999, 39:1-24
[9] A. Bellen, M. Zennaro.Strong Contractivity Properties of Numerical Methods forOrdinary and Delay Di?erential Equations. App. Numer. Math. 1992, 9: 321-346
[10] V. K. Barwell.On the Asymptotic Behavior of the Solution of Di?erential Di?er-ence Equations. Utilitas. Math. 1974, 6: 189-194
[11] H. Brunner.The numerical solutions of neutral Volterra integro-di?erential equa-tions with delay argument. Appl. Numer. Math. 1994, 1:309-322
[12] B. Cahlon.On the numerical stability of Volterra integral equations with delay ar-guments. J. Comput. Appl. Math. 1990, 33:97-104
[13] B. Cahlon.On the stability of Volterra integral equations with a lagging argument.BIT. 1995, 35:19-29
[14] C. Lubich.On the convergence of multistep methods for nonlinear sti? di?erentialequations. Numer. Math., 58(1991)839-853
[15] M.R. Crisci, E. Russo.A Vecchio stability results for one step discretized colloca-tion methods in the numerical treatment of Volterra integral equations. Math. Comp.1992, 58:119-134
[16] E. Hairer, G. Wanner.Solving ordinary di?erential equations Π Sti? anddi?erential-algebraic problems. Spinger-Verlag, Berlin, 1991
[17] E. Hairer.A- and B-stability for Runge-Kutta methods characterizations and equiv-alence. Numer. Math, 48(1986), 383-389
[18] H. Brunner, E. Hairer, S. P. Nprsett.Runge-Kutta theory for Volterra integralequations of the second kind. Math. Comp, 1982, 39: 147-163
[19] W. H. Enright, Min. Hu.Continuous Runge-Kutta methods for neutral Volterraintegro-di?erential equations with delay. Apll. Numer. Math. 1997, 24:175-190
[20] Enright W. H., Hayashi H..Convergence analysis of the solution of retarted andneutral delay di?erential equations by continuous numerical methods. SIAM. J. Nu-mer. Amal. 1998, 35:572-585
[21] N. J. Ford, C. T. H. Baker.Qualitative behaviour and stability of solutions ofdiscretised nonlinear Volterra integral equations of convolution type. J. Comp. Appl.Math. 1996, 66:213-225
[22] G. Wanner, E. Hairer, S. P. Norsett.Order star and stability theorems. BIT.18(1978): 475-489
[23] G. Dahlquist.Error analysis for a class of methods for sti? nonlinear initial valueproblems. Numer. Anal. Dundee, Lecture Notes in Math 506, Berlin: Springer-Verlag,1975, 60-74
[24] G. Dahlquist.G-stability is equivalent to A-stability. BIT, 18(1978): 384-401
[25] J. K. Hale.Theory of Functional Di?erential Equations. Springer, New York, 1977
[26] J. K. Hale.Homoclinic Orbits and Chaos in Delay Equations. In proceedings of theNinth Dundee Conference on Ordinary and Partial Di?erential Equations, edited byB. D. Sleeman et al, New York, 1986
[27] Huang Chengming.B-convergence properties of algebraically stable general linearmethods for strictly dissipative initial value problems. Proc. Int. Conf. on Comput.Di?erential Equations and Dynamical Systems, Beijing, 8-79, 1992
[28] Huang C. M., Li. S. F., Fu. H. Y..Stability and error analysis of one-leg methodsfor nonlinear delay di?erential equations. J. Comput. Appl. Math, 1999, 103:263-279
[29] C. M. Huang, S. Li, H. Fu, G. Chen.Stability Analysis of Runge-Kutta Methodsfor Nonlinear Delay Di?erential Equations. BIT. 1999, 39: 270-280
[30] C. M. Huang, S. Li, H. Fu, G. Chen.Stability and Error Analysis of One-LegMethods for Nonlinear Delay Di?erential Equations. J. Comput Appl. Math. 1999,103: 263-279
[31] C. M. Huang, Q. Chang.Linear Stability of General Linear Methods for Systemsof Neutral Delay Di?erential Equations. Appl. Math. Letters. 2001, 14: 1017-1021
[32] C. M. Huang, V. Stefan. Stability of Runge-Kutta-Pouzet methods for Volterraintegral and integro-di?erential equations with delays. Rep. TW361, K. U. Leuven.2003, 6
[33] C. Huang, S.Vandewalle. An Analysis of Delay-Depentant Stability for Ordinaryand Patial Di?erential Equations with Fixed and Distributed Delays. SIAM J. SciComput. 2004, 25(5): 1608-1632
[34] K. J. int’Hout, M. N. Spijker.The θ-methods in the Numerical Solution of DelayDi?erential Equations. Proc. Interat. Sem. NUMDIFF-5. Leipzig. 1989
[35] K. J. int’Hout, M. N. Spijker.Stability Analysis of Numerical Methods for DelayDi?erential Equations. Numer. Math. 1991, 59: 804-814
[36] K. J. int’Hout.The Stability of A Class of Runge-Kutta Methods for Delay Di?er-ential Equations. Appl. Numer. Math. 1992, 9: 347-355
[37] K. J. int’Hout, M. N. Spijker.A New Interpolation Procedure for AdaptingRunge-Kutta Methods for Delay Di?erential Equations. BIT. 1992, 32: 643-649
[38] K. J. int’Hout.The Stability of the θ-methods for Systems of Delay Di?erentialEquations. Report Series. Univ. Auckland. 1993, No.282
[39] K. J. int’Hout.A Note on Unconditional Maximum Norn Contrativity of Diago-nally Split Runge-Kutta Methods. SIAM. J. Numer. Anal. 1996, 33: 1125-1134
[40] K. J. int’Hout, M. N. Spijker.Stability Analysis of Runge-Kutta Methods forSystems of Delay Di?erential Equations. IMA. J. Numer. Anal. 1997, 17: 17-27
[41] G. D. Hu, T. Mitsui.Stability Analysis of Numerical Methods for Systems of Neu-tral Delay-Di?erential Equations. BIT. 1995, 35: 504-515
[42] G. D. Hu, B. Cahlon.Numerically Stable Step-Size for Neutral Delay Di?erentialSystems with Multiple Delays. J. Comput Appl. Math. 1999, 102: 221-234
[43] Z. Jackiewicz.Asymptotic Stability Analysis of θ-Methods for Functional Di?eren-tial Equations. Numer. Math. 1984, 43: 389-396
[44] J. C. Butcher.A stability property of implicit Runge-Kutta methods. BIT, 15(1975):358-361
[45] J. C. Butcher.Linear and nonlinear stability for general linear methods. BIT.27(1987): 182-189
[46] J. D. Lambert.Computational methods in ordinary di?erential equations. John Wi-ley and Sons Ltd, 1973
[47] L. Kenneth, K. L. Cooker, W. Joseph.Retarded Di?erential Equations withPiecewise Constant Delays. J. Math. Anal. and Appl. 1984, 99: 265-297
[48] K. Burrage, J. C. Butcher.Stability criteria for implicit Runge-Kutta methods.SIAM. J. Numer. Anal. 16(1979): 46-57
[49] K. Dekker, J. G. Verwer.Stability of Runge-Kutta Methods for Sti? NonlinearDi?erential Equations. North-Holland, Amsterdam, 1984
[50] K. Strehmel, R. Weiner, I. Dannehl.On error behavior of partitioned lin-early implicit Runge-Kutta methods for sti? and di?erential algebraic systems. BIT,30(1990), 358-375
[51] T. Koto.A Stability Property of A-stable natural Runge-Kutta methods for Systemsof Delay Di?erential Equations. BIT. 1994, 34: 262-267
[52] T. Koto.Stability of Runge-Kutta methods for delay integro-di?erential equations.J. Comp. Appl. Math. 2002, 45: 483-495
[53] J. X. Kuang, H. J. Tian.The Asymptotic Behaviour of Theoretical and Numericalfor the De?erential Equations with Several Delay Terms. J. Shanghai Normal Univ.1994, 23: 1-10
[54] J. X. Kuang, H. J. Tian.The Asymptotic Analysis of One-Parameter Methods forNeutral Delay Di?erential Equations. BIT. 1994, 34: 400-408
[55] L. F. Shampine, Gear C. W.A user’s view of solving sti? ordinary di?erentialequations. SIAM Review. 1979, 21:1-17
[56] Li Shoufu.B-convergence of general linear methods. Proc. Bail-V. Int.Conf(Shanghai, 1988), 203-208
[57] Li Shoufu.Stability and B-convergence of general linear methods. J. Comput. Appl.Math, 28(1989), 565-575
[58] Li Shoufu.Nonlinear stability of general linear methods. J. Comput. Math,9(2)(1991), 97-104
[59] Li Shoufu.B-convergence of general multivalue methods applied to sti? problems inBanach spaces. Science In China(Series A), 36(1993), 1-13
[60] Li Shoufu.B-convergence properties of multistep Runge-Kutta methods. Math. Com-put. 62(1994), 565-575
[61] Li Shoufu.Theory of computational methods for sti? di?erential equations. HunanSci. and Tech. Press, 1997(in Chinese)
[62] M. Z. Liu, M. N. Spiijker.The Stability of the θ-methods in the Numerical Solutionof Delay Di?erential Equations. IMA. J. Numer. Anal. 1990, 10: 31-48
[63] J. Mori, E. Noldus, M. Kuwahara.A Way to Stabilize Linear Systems withDelayes State. Automatica. 1983, 19:571-573
[64] Y. S. Chin.On the Equivalence Problem of Di?erential Equations and Di?erenceEquations in the Theory of Stability. Sci. Record. 1957, 1:287-289
[65] Y. S. Chin et al.E?ect of Time-lags on Stability of Dynamical Systems. Sci. Sinica.1960, 9:719-747
[66] L. Qiu et al.The NGP-Stability of Runge-Kutta Methods for Systems of NeutralDelay Di?erential Equations. Numer. Math. 1999, 81: 451-459
[67] R. Frank, J. Schneid, C. W. Ueberhuber.The concept of B-convergence. SIAM.J. Numer. Anal, 18(1981), 753-780
[68] R. Frank, J. Schneid, C. W. Ueberhuber.Order results for implicit Runge-Kutta methods applied to sti? systems. SIAM. J. Numer. Anal, 22(1985), 515-534
[69] R. Frank, J. Schneid, C. W. Ueberhuber.Stability properties of implicit Runge-Kutta methods. SIAM. J. Numer. Anal, 22(1985), 497-515
[70] R. Frank, J. Schneid, H. Lehner.B-convergence properties of defect correctionmethods. |,Π. Numer. Math, 46(1986), 139-188
[71] S. Schneider.Convergence results for general linear methods on singular perturba-tion problems. BIT. 33(1993), 670-686
[72] Siqing Gan, Geng Sun, Weimin Zheng.Errors of Linear Methods and Runge-Kutta Methods for Singular Perturbation Problems with Delays. J. Comput. Appl.Math., 44(2002)1157-1173
[73] H. J. Tian, J. X. Kuang.The Stability of Linear Multistep Methods for Systemsof Delay Di?erential Equations. Numer. Math. A. Journal of Chinese Univ. 1995, 4:10-16
[74] H. J. Tian, J. X. Kuang.The numerical stability analysis of Runge-Kutta Methodsfor Volterra integral equations with delay terms. J. AMM. 1995, 16: 485-491
[75] H. J. Tian, J. X. Kuang.The Stability Analysis of the θ-methods for Delay Dif-ferential Equations. J. Comput. Math. 1996, 14: 203-212
[76] H. J. Tian, J. X. Kuang.The Numerical Stability of Linear Multistep Methodsfor Delay Di?erential Equations with Several Delays. SIAM. J. Numer. Anal. 1996,33(3): 883-889
[77] H. Tian.Numerical treatment of singularly perturbed delay di?erential equations.Ph.D. thesis, University of Manchester(2000).
[78] Hongjiong Tian.The exponential asymptotic stability of singularly perturbed delaydi?erential equations with a bounded lag. J. Math. Anal. Appl. 270(2002)143-149
[79] C.T.H. Baker, C.A.H. Paul, H. Tian.Di?erential algebraic equations with after-e?ect. J. Comput. Appl. Math. 140(2002):63-80
[80] L. Torelli.Stability of Numerical Methods for Delay De?erential Equations. J.Comput. Appl. Math. 1989, 25: 15-26
[81] L. Torelli. Stability of Numerical Methods for Delay Di?erential Equations Acti-vated Tanks. J. Comput. Appl. Math. 1989, 25: 15-26
[82] R. Verminglio, L. Torelli. A Stable Numerical Approach for Implicit Non-Linearneutral Delay Di?erential Equations. BIT. 2003, 43: 195-215
[83] R. Vermiglio.On the stability of Runge-Kutta methods for delay integral equations.Numer. Math. 1992, 61: 561-577
[84] J. Wiener.Retarded Di?erential Equations with Piecewise Constant Delays. J. Math.Anal. Appl. 1984, 99: 265-297
[85] D. S. Watanable, M. G. Roth.The Stability of Di?erence Formulas for DelayDi?erential Equations. SIAM. J. Numer. Anal. 1985, 22: 132-145
[86] W. Auzinger, R. Frank, G. Kirlinger.Extending convergence theory for non-linear sti? problems. Part1. BIT. 36:4, 1996, 635-652
[87] Xiao Aiguo.An algebraic characterization of B-convergent general Linear methodsfor strictly dissipative initial value problems. Proc. 3rd. Int. Confon. Syst. Simul. Sci.Comput, Bejing Int. Acad. Pub. 135-137, 1995
[88] Xiao Aiguo, Li Shoufu, Fu hongyuan, Chen Guangnan.Convergence ofLinear multistep methods for two-parameter singular perturbation problems. ACTAMATHEMATICAE APPLICATAE SINICA, Vol 17, No.2, 2001
[89] B. Yang, L. Qiu, T. Mitsui.GPk-Stability of Runge-Kutta Methods for GeneralizedDelay Di?erential Systems. Comput. Math. Appl. 1999, 37: 89-97
[90] M. Zennaro.P-Stability Properties of Runge-Kutta Methods for delay Di?erentialEquations. Numer. Math. 1986, 49: 302-318
[91] M. Zennaro.Contracticity of Runge-Kutta Methods with Respect to Forcing Terms.Appl. Numer. Math. 1993, 10: 321-345
[92] Zhang C. J., Zhou. S. Z.Nonlinear stability and D-convergence of Runge-Kuttamethods for delay di?erential equations. J. Comput. Appl. Math. 1997, 85:225-237
[93] C. J. Zhang, X. X. Liao, W. Cheng.GPk-stability of the Multistep Runge-KuttaMethods for a Class of Multidelay Systems. Math. Appl. 2000, 13(3): 139-142
[94] C. J. Zhang, X. X. Liao. Stability of BDF methods for Volterra integral equationswith delay. Comp. Math. Appl. 2002, 43:95-102
[95] C. Zhang, S. Vandewalle. Stability Analysis of Volterra Delay-Integro-Di?erential Equations and Their Backward Di?erentiation Time Discretization. J.Comput. App. Math. 2004, 164/165: 797-814
[96] C. J. Zhang, V. Stefan. Stability analysis of Runge-Kutta methods for nonlinearVolterra delay di?erential equations. IMA. J. Numer. Anal. 2004, 24: 193-214
[97] 李寿佛. Banach空间中非线性刚性Volterra泛函微分方程稳定性分析. 中国科学.2005, 35(3), 286-301
[98] 侯文星.一类奇异摄动初值问题BDF方法的收敛性. 湘南学院学报,2004. Vol.25No.5:35-36
[99] 徐阳,赵景军,刘明珠. 二阶延迟微分方程θ -方法的TH-稳定性. 计算数学. 2004,26(2), 189-192
[100] 王晚生,李寿佛. 非线性中立型延迟微分方程稳定性分析. 计算数学. 2004, 26(3),303-314
[101] 郭长勇. 几类延迟微分方程的数值稳定性. 哈尔滨工业大学硕士论文. 2002
[102] 甘四清,孙耿.时滞奇异摄动问题单支方法的收敛性. 中国科学(A辑). 2001,31(4): 314-323
[103] 张诚坚, 高健.隐式Euler方法关于V olterra延迟积分方程的数值稳定性. 应用数学,2000, 13(4):130-132
[104] 范振成.二阶延迟微分方程数值稳定性分析. 哈尔滨工业大学硕士论文. 2000
[105] W. Cucas.微分方程模型. 国防科技大学出版社. 1999
[106] 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(A辑). 1998, 28(8): 713-720
[107] 李寿佛.刚性微分方程算法理论. 长沙:湖南科学技术出版社,1997
[108] 刘德贵,费景高,韩天敏.刚性大系统数字仿真方法. 河南科学技术出版社,1996
[109] 秦元勋等.带有时滞的动力系统的运动稳定性. 科学出版社. 1992
[110] 刘永清.滞后大系统的工程实际与稳定性方法. 科学出版社. 1992
[111] 钱学森.工程控制论. 科学出版社. 1958
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