关于几类二阶延迟微分方程数值解及其稳定性的研究
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摘要
本文主要研究二阶延迟微分方程的数值解及其稳定性。全文由四章组成。
第一章主要介绍了延迟微分方程的研究背景以及课题的现实意义。
第二章主要讨论了二阶延迟微分方程周期解的存在唯一性及数值解法。首先,根据一个引理给出并且证明了方程存在唯一周期解的充分条件,然后利用牛顿法研究了周期数值解。
第三章主要讨论了二阶滞后型微分方程的理论解解和数值解的稳定性。本章主要包括两个方面：一方面,由二阶延迟微分方程的特征方程,给出其渐近稳定的充要条件；另一方面,给出数值解的单支θ-方法的稳定性质,证明了当θ=1时数值解是稳定的。
第四章主要讨论了中立型方程的理论解和数值解的稳定性。首先,利用特征根分析方法,获得了理论解稳定的充要条件；其次,在理论解稳定的基础之上,考虑方程单支θ-方法的稳定性质,证明当θ=1时,单支θ-方法是稳定的。
This paper is concerned with the numerical and stability of solutions for several second-order delay differential equations, which is composed of four parts.
In the first chapter, the research background and practical significance of the topics are simply summarized.
The second chapter is concerned with the numerical solutions and stability of second-order delay differential equations. The sufficient and necessary condition is given under which the existence and uniqueness of periodic solutions is guaranteed. The numerical solutions and its stability are considered by using Newton method.
In the third chapter, the analytical solutions and numerical solutions of second-order delay differential equation are considered by one-legθ-methods. A necessary and sufficient condition is given by analyzing eigenvalues of characteristic equation. It is proved that the numerical method is stable whenθ=1.
The four chapter, we deal with the neutral equation as follow The eigenvalues of linear characteristic equation are analyzed in detail. The numerical solutions by one-legθ- methods are constructed and its stability is obtained. Specially, it is proved that the numerical method is stable whenθ=1.
第一章主要介绍了延迟微分方程的研究背景以及课题的现实意义。
第二章主要讨论了二阶延迟微分方程周期解的存在唯一性及数值解法。首先,根据一个引理给出并且证明了方程存在唯一周期解的充分条件,然后利用牛顿法研究了周期数值解。
第三章主要讨论了二阶滞后型微分方程的理论解解和数值解的稳定性。本章主要包括两个方面：一方面,由二阶延迟微分方程的特征方程,给出其渐近稳定的充要条件；另一方面,给出数值解的单支θ-方法的稳定性质,证明了当θ=1时数值解是稳定的。
第四章主要讨论了中立型方程的理论解和数值解的稳定性。首先,利用特征根分析方法,获得了理论解稳定的充要条件；其次,在理论解稳定的基础之上,考虑方程单支θ-方法的稳定性质,证明当θ=1时,单支θ-方法是稳定的。
This paper is concerned with the numerical and stability of solutions for several second-order delay differential equations, which is composed of four parts.
In the first chapter, the research background and practical significance of the topics are simply summarized.
The second chapter is concerned with the numerical solutions and stability of second-order delay differential equations. The sufficient and necessary condition is given under which the existence and uniqueness of periodic solutions is guaranteed. The numerical solutions and its stability are considered by using Newton method.
In the third chapter, the analytical solutions and numerical solutions of second-order delay differential equation are considered by one-legθ-methods. A necessary and sufficient condition is given by analyzing eigenvalues of characteristic equation. It is proved that the numerical method is stable whenθ=1.
The four chapter, we deal with the neutral equation as follow The eigenvalues of linear characteristic equation are analyzed in detail. The numerical solutions by one-legθ- methods are constructed and its stability is obtained. Specially, it is proved that the numerical method is stable whenθ=1.
引文
[1]郑祖庥.泛函微分方程理论,安徽教育出版社[M].1993.
[2]蒋威.退化、时滞微分系统,安徽大学出版社[M].1998.
[3]Z. Jackiewicz. Asymptotic stability analysis of θ-methods for functional differential equations[J]. Numer, Math.43(1984),389-396.
[4]刘永清.滞后大系统的工程实际与稳定性方法[M].北京,科学出版社.
[5]J. K. HALE. Homoclinic orbits and chaos in Delay Equations, In B. D. Sleeman etal. Proceedings of the Ninth Dundee conference on ordinary and Partial Differential Equations[M]. New York,1986.
[6]H.S.Tsien. Engineer Cybernetics[M].1945.
[7]Xiao D., Li W. Stability and bifurcation in a delayed rato-dependent predetor-prey system [J]. Proc. Edinburgh Math. Soc.,2003,45:205-220.
[8]应科俊,温朝辉.一类中立型捕食系统的hopf分支[J].山东理工大学学报,2009,23：39-41.
[9]W.CUCAS.微分方程[M].长沙,国防科技大学出版社,1999.
[10]徐阳,赵景军,刘明珠.二阶延迟微分方程θ-方法的TH-稳定性[J].计算数学,2004,26(3)：189-192.
[11]葛淑君.二阶延迟微分方程θ-方法数值解稳定性[J].哈尔滨师范大学自然科学学报,2006,22：29-30.
[12]王丽,张传义.带逐段常变量的二阶中立型延迟微分方程的概周期解[J].数学学报,2010,53：227-232.
[13]G. L. Marchuk. Mathematical models in Immunology. Numerical Methods and Experiments [M]. Thirdrev.& enl. ed NAUKA:Moscow 1991 (in Russian).
[14]秦元勋,刘永清,王联,郑祖庥.带有时滞的动力系统的运动稳定性[M].科学出版社,1992.
[15]C. T. H. Baker, G. A. Bocharov, F. A. Rihan. A Report on the use of Delay Differential Equations in Numerical Modelling in the Biosciences[J]. MCCM Technical Report 343 ISSN 1999:1360-1725.
[16]Y. S. CHIN. On the Equivalence Problem of Differential Equations and Difference Equations in the Theory of Stability[J]. Sci. Record.1957, 1:287-289.
[17]Y. S. CHIN, ETAL. Effect of Time-lag on Stability of Dynamical Systems [J]. Sci.Sinica.1960,9:719-747.
[18]-, Natural continuous Extensions of Runge-Kutta Methods[J]. Math. Comp.1986,46:119-133.
[19]R. Bellman, K. L. Cooke. Differential-Difference Equations [M]. Academ-ic Press New York. London 1963,444-446.
[20]V. K. BARWELL. On the Asymptotic Behaviour of the Solution of Differ-encial Difference Equations[J]. Utilitas. Math.1974,6:189-194.
[21]D. S. Watanade. The Stability of Difference Formulas for Delay Differential Equations[J]. SIAM. J.Numer. Anal.1985,22:132-145.
[22]L. Torelli. Stability of Numerical methods for Delay Differential Equations[J]. J. Comput. Appl. Math.1989,25:15-26.
[23]Kuang J. X., Tian H. J. The Asymptotic Behaviour of Theoretical and Numerical for the Differential Equations with several Delay Terms [J]. J. Shanghai Normal Univ.1994,23:1-10.
[24]Kuang J. X., Tian H. J. The Asymptotic Analysis of One-Parameter Methods for Neutral Delay Differential Equations[J]. BIT.1994, 34:400-408.
[25]张诚坚,周树立.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[J].中国科学(A辑).1998,28(8)：713-720.
[26]Gan Siqing, Zheng Weiming. Stability of Multistep Runge-Kutta Methods for Systems of Functional-Differential and Functional Equations[J]. Appl. Math. Letter.2004,17:585-590.
[27]Gan Siqing. Asymptotic Stability of Rosenbrock Methods for Systems of Functional Differential and functional Equations[J]. Math. Compt. Modelling.2006,44:144-150.
[28]R. Bellman, K. L. Cooke. Differential-Difference Equations[M]. Academic Press New York. London 1963,450-452.
[29]K. L. Cooke, Z. Grossman. Discrete Delay, Distributed Delay and Stabil-ity Switches [J]. J. Math. Anal. Appl.1982,86:592-627.
[30]Stepan. Retarded Dynamical Systems, Stability and Characteristic Functions[M], Long-man Scientific and Technical U.K.1989.
[31]郭长勇,赵景军,刘明珠.二阶延迟微分方程解析解的渐近稳定性[J],黑龙江大学自然科学学报.2002,19：5-7.
[32]黄乘明,李文皓.一类二阶延迟微分方程梯形方法的延迟依赖稳定性分析[J].2007,29：155-162.
[33]徐阳,赵景军.Volterra型时滞积分方程单支θ-方法[J],数学物理学报,2008,28A(5)：942-944.
[34]张正球,庾建设.一类时滞Duffing型方程周期解[J].高校应用数学学报A辑,1998,13(4)：389-392.
[35]马世旺,瘐建设,王志成.拟线性泛函微分方程周期解的存在唯一性[J].数学年刊,2001,22(A)：105-110.
[36]许敏.非线性微分方程边值问题周期解的研究[D].南京：南京信息工程大学理学院数学系,2006.
[37]徐庆.用微分连续法求解Duffing型方程的2π周期解[J].吉林大学自然科学学报,1990(3)：1-6.
[38]林福荣,杨丽伟.一类时滞Duffing型方程周期解的存在唯一性及数值解法[J].汕头大学学报,2008,23(3)：14-19.
[39]唐美兰,刘新歌,郭水霞.时滞Duffing型微分方程周期解的存在唯一性[J].纯粹数学与应用数学,2005,21(2)：99-102.
[40]关治,陆金甫.数值分析基础[M].北京,高等教育出版社,1998,351-353.
[41]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].北京：科学出版社,1987.
[42]范振成.二阶延迟微分方程数值稳定性研究[的].哈尔滨工业大学硕士学位论文,2000,7.
[43]R. Bellman, K. L. Cooke. Differential-difference Equations[M]. New York:Academic Press,1963.
[44]W. L. Miranker. Existence, uniqueness and stability of solutions of systems nonlinear difference differential equations[J]. J. Mech., 11(1962)
[45]Y.K.Liu. Stability analysis of θ-methods for neutral functional-differential equations[J]. Numer Math,1995,70:473-485.
[46]V. K. BARWELL. Special stability problem for functional differential equations[J]. BIT,1975,15:132-135.
[47]M. Z. LIU, M. N. SPIJKER. Stability of the θ-methods in numerical solution of delay differential equations[J]. IMAJ Numer Anal,1990, 10:31-48.
[48]K. J. in't HOUT. Stability of Runge-Kutta methods for systems of delay differential equations[J]. IMAJ Numer Anal,17:17-27.
[49]TIAN H J, KUANG J X. Stability of the θ-methods in numerical solution of delay differential equations with several delay terms [J]. J Comput Appl Math,1995,58:171-181.
[50]R.Bellman, K. L. Cooke. Differential-difference Equations[M]. New York:Academic Press,1963,443-444.
[51]G. PAPAGEORGIOU, I TH FAMELIS. On using explicit Runge-Kutta-Nysyrom methods for the treatment of retarded differential equations with periodic solutions[J]. Appl Math Comput,1999,102:63-76.
[2]蒋威.退化、时滞微分系统,安徽大学出版社[M].1998.
[3]Z. Jackiewicz. Asymptotic stability analysis of θ-methods for functional differential equations[J]. Numer, Math.43(1984),389-396.
[4]刘永清.滞后大系统的工程实际与稳定性方法[M].北京,科学出版社.
[5]J. K. HALE. Homoclinic orbits and chaos in Delay Equations, In B. D. Sleeman etal. Proceedings of the Ninth Dundee conference on ordinary and Partial Differential Equations[M]. New York,1986.
[6]H.S.Tsien. Engineer Cybernetics[M].1945.
[7]Xiao D., Li W. Stability and bifurcation in a delayed rato-dependent predetor-prey system [J]. Proc. Edinburgh Math. Soc.,2003,45:205-220.
[8]应科俊,温朝辉.一类中立型捕食系统的hopf分支[J].山东理工大学学报,2009,23：39-41.
[9]W.CUCAS.微分方程[M].长沙,国防科技大学出版社,1999.
[10]徐阳,赵景军,刘明珠.二阶延迟微分方程θ-方法的TH-稳定性[J].计算数学,2004,26(3)：189-192.
[11]葛淑君.二阶延迟微分方程θ-方法数值解稳定性[J].哈尔滨师范大学自然科学学报,2006,22：29-30.
[12]王丽,张传义.带逐段常变量的二阶中立型延迟微分方程的概周期解[J].数学学报,2010,53：227-232.
[13]G. L. Marchuk. Mathematical models in Immunology. Numerical Methods and Experiments [M]. Thirdrev.& enl. ed NAUKA:Moscow 1991 (in Russian).
[14]秦元勋,刘永清,王联,郑祖庥.带有时滞的动力系统的运动稳定性[M].科学出版社,1992.
[15]C. T. H. Baker, G. A. Bocharov, F. A. Rihan. A Report on the use of Delay Differential Equations in Numerical Modelling in the Biosciences[J]. MCCM Technical Report 343 ISSN 1999:1360-1725.
[16]Y. S. CHIN. On the Equivalence Problem of Differential Equations and Difference Equations in the Theory of Stability[J]. Sci. Record.1957, 1:287-289.
[17]Y. S. CHIN, ETAL. Effect of Time-lag on Stability of Dynamical Systems [J]. Sci.Sinica.1960,9:719-747.
[18]-, Natural continuous Extensions of Runge-Kutta Methods[J]. Math. Comp.1986,46:119-133.
[19]R. Bellman, K. L. Cooke. Differential-Difference Equations [M]. Academ-ic Press New York. London 1963,444-446.
[20]V. K. BARWELL. On the Asymptotic Behaviour of the Solution of Differ-encial Difference Equations[J]. Utilitas. Math.1974,6:189-194.
[21]D. S. Watanade. The Stability of Difference Formulas for Delay Differential Equations[J]. SIAM. J.Numer. Anal.1985,22:132-145.
[22]L. Torelli. Stability of Numerical methods for Delay Differential Equations[J]. J. Comput. Appl. Math.1989,25:15-26.
[23]Kuang J. X., Tian H. J. The Asymptotic Behaviour of Theoretical and Numerical for the Differential Equations with several Delay Terms [J]. J. Shanghai Normal Univ.1994,23:1-10.
[24]Kuang J. X., Tian H. J. The Asymptotic Analysis of One-Parameter Methods for Neutral Delay Differential Equations[J]. BIT.1994, 34:400-408.
[25]张诚坚,周树立.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[J].中国科学(A辑).1998,28(8)：713-720.
[26]Gan Siqing, Zheng Weiming. Stability of Multistep Runge-Kutta Methods for Systems of Functional-Differential and Functional Equations[J]. Appl. Math. Letter.2004,17:585-590.
[27]Gan Siqing. Asymptotic Stability of Rosenbrock Methods for Systems of Functional Differential and functional Equations[J]. Math. Compt. Modelling.2006,44:144-150.
[28]R. Bellman, K. L. Cooke. Differential-Difference Equations[M]. Academic Press New York. London 1963,450-452.
[29]K. L. Cooke, Z. Grossman. Discrete Delay, Distributed Delay and Stabil-ity Switches [J]. J. Math. Anal. Appl.1982,86:592-627.
[30]Stepan. Retarded Dynamical Systems, Stability and Characteristic Functions[M], Long-man Scientific and Technical U.K.1989.
[31]郭长勇,赵景军,刘明珠.二阶延迟微分方程解析解的渐近稳定性[J],黑龙江大学自然科学学报.2002,19：5-7.
[32]黄乘明,李文皓.一类二阶延迟微分方程梯形方法的延迟依赖稳定性分析[J].2007,29：155-162.
[33]徐阳,赵景军.Volterra型时滞积分方程单支θ-方法[J],数学物理学报,2008,28A(5)：942-944.
[34]张正球,庾建设.一类时滞Duffing型方程周期解[J].高校应用数学学报A辑,1998,13(4)：389-392.
[35]马世旺,瘐建设,王志成.拟线性泛函微分方程周期解的存在唯一性[J].数学年刊,2001,22(A)：105-110.
[36]许敏.非线性微分方程边值问题周期解的研究[D].南京：南京信息工程大学理学院数学系,2006.
[37]徐庆.用微分连续法求解Duffing型方程的2π周期解[J].吉林大学自然科学学报,1990(3)：1-6.
[38]林福荣,杨丽伟.一类时滞Duffing型方程周期解的存在唯一性及数值解法[J].汕头大学学报,2008,23(3)：14-19.
[39]唐美兰,刘新歌,郭水霞.时滞Duffing型微分方程周期解的存在唯一性[J].纯粹数学与应用数学,2005,21(2)：99-102.
[40]关治,陆金甫.数值分析基础[M].北京,高等教育出版社,1998,351-353.
[41]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].北京：科学出版社,1987.
[42]范振成.二阶延迟微分方程数值稳定性研究[的].哈尔滨工业大学硕士学位论文,2000,7.
[43]R. Bellman, K. L. Cooke. Differential-difference Equations[M]. New York:Academic Press,1963.
[44]W. L. Miranker. Existence, uniqueness and stability of solutions of systems nonlinear difference differential equations[J]. J. Mech., 11(1962)
[45]Y.K.Liu. Stability analysis of θ-methods for neutral functional-differential equations[J]. Numer Math,1995,70:473-485.
[46]V. K. BARWELL. Special stability problem for functional differential equations[J]. BIT,1975,15:132-135.
[47]M. Z. LIU, M. N. SPIJKER. Stability of the θ-methods in numerical solution of delay differential equations[J]. IMAJ Numer Anal,1990, 10:31-48.
[48]K. J. in't HOUT. Stability of Runge-Kutta methods for systems of delay differential equations[J]. IMAJ Numer Anal,17:17-27.
[49]TIAN H J, KUANG J X. Stability of the θ-methods in numerical solution of delay differential equations with several delay terms [J]. J Comput Appl Math,1995,58:171-181.
[50]R.Bellman, K. L. Cooke. Differential-difference Equations[M]. New York:Academic Press,1963,443-444.
[51]G. PAPAGEORGIOU, I TH FAMELIS. On using explicit Runge-Kutta-Nysyrom methods for the treatment of retarded differential equations with periodic solutions[J]. Appl Math Comput,1999,102:63-76.