分段连续微分方程数值方法的研究
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摘要
本文主要研究了自变量分段连续微分方程数值解的稳定性和振动性。
第一章,概要地介绍了自变量分段连续时滞微分方程的研究目的和意义,以及近些年来在该领域的研究现状。这类方程广泛存在于应用科学的各个领域中,其数值解的稳定性分析具有重要的理论价值和实际意义。振动性的研究是时滞微分方程定性理论研究的一个重要方面,而时滞微分方程又可以刻画生活中的许多模型,因此,研究时滞微分方程的振动性是很有必要的。
第二章,针对分段连续时滞微分方程构造出Rosenbrock方法的计算格式,并将该方法应用于具体的分段连续时滞微分方程,讨论得到数值解渐近稳定的充分条件,同时考虑解析解的渐近稳定性条件,加以改进,得到最后的结论。然后利用特征方程的方法对Rosenbrock方法进行分析,最后得到Rosenbrock方法保持数值解振动性的充要条件。
第三章,将分离变量法应用于分段连续抛物型微分方程,进而得到解析解的稳定性条件,在此条件下,针对具体的纯量抛物微分方程的Crank-Nicholson差分格式进行分析,得出关于方程参数的稳定性条件。对于每一部分的讨论,都给出了相应的数值算例,这些算例验证了理论上推得的结论的正确性。
This paper deals with the numerical stability of the differential equations with piecewise continuous arguments mainly.
First, this paper presents the purpose and the significance of researching about the delay differential equations with piecewise continuous arguments in brief, what’s more, it introduces the research actuality in this field. This class of equations appears in many fields of applied sciences, and the stability analysis of the numerical solution has important theoretical and practical significance. The research about oscillation is an important part of the research about the differential equations. Also, the delay differential equations can show many phenomena, so it is necessary to research about the oscillation of the delay differential equations.
Secondly, it applies the Rosenbrock method to the delay differential equations with piecewise continuous arguments, then discusses the stability conditions of numerical solution when Rosenbrock method is applied to the concrete delay differential equations with piecewise continuous arguments. Also, the preservation of oscillatory property of Rosenbrock method is considered.
Last but not least, we apply the Separation of variables method to the parabolic partial differential equation with piecewise continuous arguments, and obtain the stability condition of the concrete scalar parabolic partial equation using Crank-Nicholson scheme, and the stability condition is about the parameters of the equation.
Moreover, the relevant numerical examples are given in every part. These experiments verify the conclusion obtained in the theoretical analysis.
第一章,概要地介绍了自变量分段连续时滞微分方程的研究目的和意义,以及近些年来在该领域的研究现状。这类方程广泛存在于应用科学的各个领域中,其数值解的稳定性分析具有重要的理论价值和实际意义。振动性的研究是时滞微分方程定性理论研究的一个重要方面,而时滞微分方程又可以刻画生活中的许多模型,因此,研究时滞微分方程的振动性是很有必要的。
第二章,针对分段连续时滞微分方程构造出Rosenbrock方法的计算格式,并将该方法应用于具体的分段连续时滞微分方程,讨论得到数值解渐近稳定的充分条件,同时考虑解析解的渐近稳定性条件,加以改进,得到最后的结论。然后利用特征方程的方法对Rosenbrock方法进行分析,最后得到Rosenbrock方法保持数值解振动性的充要条件。
第三章,将分离变量法应用于分段连续抛物型微分方程,进而得到解析解的稳定性条件,在此条件下,针对具体的纯量抛物微分方程的Crank-Nicholson差分格式进行分析,得出关于方程参数的稳定性条件。对于每一部分的讨论,都给出了相应的数值算例,这些算例验证了理论上推得的结论的正确性。
This paper deals with the numerical stability of the differential equations with piecewise continuous arguments mainly.
First, this paper presents the purpose and the significance of researching about the delay differential equations with piecewise continuous arguments in brief, what’s more, it introduces the research actuality in this field. This class of equations appears in many fields of applied sciences, and the stability analysis of the numerical solution has important theoretical and practical significance. The research about oscillation is an important part of the research about the differential equations. Also, the delay differential equations can show many phenomena, so it is necessary to research about the oscillation of the delay differential equations.
Secondly, it applies the Rosenbrock method to the delay differential equations with piecewise continuous arguments, then discusses the stability conditions of numerical solution when Rosenbrock method is applied to the concrete delay differential equations with piecewise continuous arguments. Also, the preservation of oscillatory property of Rosenbrock method is considered.
Last but not least, we apply the Separation of variables method to the parabolic partial differential equation with piecewise continuous arguments, and obtain the stability condition of the concrete scalar parabolic partial equation using Crank-Nicholson scheme, and the stability condition is about the parameters of the equation.
Moreover, the relevant numerical examples are given in every part. These experiments verify the conclusion obtained in the theoretical analysis.
引文
1 R. Bellman, K. L. Cooke. Differential-Difference Equations. Academic Press. New York, 1963:35~141
2 V. B. Kolmanovskii, V. R. Nosov. Stability of Functional Differential Equations. Academic Press. New York, 1986:25~163
3 J. Wiener. Generalized Solutions of Functional Differential Equations. World Scientific. Singapore. 1993:1~212
4 E. A. Grove, I. Gy o??ri and G. Ladas. On the Characteristic Equations for Equations with Continuous and Piecewise Constant Delays. Rad. Mat. 1989, 5(2):271~281
5 K. N. Jayasree, S. G. Deo. Variation of Parameters Formula for the Equation of Cooke and Wiener. Proc. Amer. Math. Soc. 1991, 112(1):75~80
6 K. L. Cooke, J. Wiener. Neutral Differential Equation with Piecewise Constant Argument. Boll. Un. Mat. Ital. 1987, 7(1):321~346
7 W. J. Lv, Z. W. Yang and M. Z. Liu. Stability of the Euler-Maclaurin Methods for Neutral Differential Equations with Piecewise Continuous Arguments. Appl. Math. Comput. 2007, 186(2):1480~1487
8 W. J. Lv, Z. W. Yang and M. Z. Liu. Stability of Runge-Kutta Methods for the Alternately Advanced and Retarded Differential Equations with Piecewise Continuous Arguments. Comput. Math. Appl. 2007, 54(3):326~335
9 M. Z. Liu, S. F. Ma and Z. W. Yang. Stability Analysis of Runge-Kutta Methods for Unbounded Retarded Differential Equations with Piecewise Continuous Arguments. Appl. Math. Comput. 2007, 191(1):57~66
10 Z. W. Yang, M. Z. Liu and M. H. Song. Stability of Runge-Kutta Methods in the Numerical Solution of Equation u′( t ) =au(t ) +a0 u([ t ]) +a1u([ t ?1]). Appl. Math. Comput. 2005, 162(1):37~50
11 M. Z. Liu, M. H. Song and Z. W. Yang. Stability of Runge-Kutta Methods in the Numerical Solution of Equation u′( t ) =au(t ) +a0u([ t ]). J. Comput. Appl. Math. 2004, 166(2):361~370
12 M. H. Song, Z. W. Yang and M. Z. Liu. Stability ofθ-Methods for Advanced Differential Equations with Piecewise Continuous Arguments. Comput. Math. Appl. 2005, 49(9-10):1295~1301
13 N. Guglielmi, M. Zennaro. On the Zero-stability of Variable Stepsize Multistep Methods: the Spectral Radius Approach. Numer. Math. 2001, 88(3):445~458
14 K. J. in’t Hout. On the Stability of Adaptations of Runge-Kutta Methods to Systems of Delay Differential Equations. Appl. Num. Math. 1996, 22(1),237-250
15 T. Koto. Stability ofθ-methods for Delay Integro-differential Equations. J. Comput. Appl. Math. 2003, 161(2):393~404
16 K. L. Cooke, J. Wiener. Stability Regions for Linear Equations with Piecewise Continuous Arguments. Comput. Math. Appl. 1986, 12(6):695~701
17吕万金,刘明珠.方程x′( t ) =a x (t ) + a2 x ([t + 2])的线性θ-方法数值稳定性.黑龙江大学自然科学学报. 2006, 23(5):700~702
18 S. M. Shah, J. Wiener. Advanced Differential Equations with Piecewise Constant Argument Deviations. Int. J. Math. Soc. 1983, 6(4):671~703
19 L. Torelli. A Sufficient Condition for GPN-Stability for Delay Differential Equations. Numer. Math. 1991, 59(1):311~320
20 G. Ladas. Y. G. Sficas and I. P. Stavroulakis. Necessary and Sufficient Conditions for Oscillations. AM. Math. mon. 1983, 90(9):637~640
21 I. Kubiaczyk, S. H. Saker. Oscillation and Stability in Nonlinear Delay Differential Equations of Population Dynamics. Math. Comput. Model. 2002, 35(3-4):295~301
22 L. Berezansky, Y. Domshlak and E. Braverman. On Oscillation Properties of Delay Differential Equations with Positive and Negative Coefficients. J. Math. Anal. Appl. 2002, 274(1):81~101
23 I. Kubiaczyk, S. H. Saker and J. Morchalo. New Oscillation Criteria for First Order Nonlinear Neutral Delay Differential Equations. Appl. Math. Comput. 2003, 142(2-3):225~242
24 H. E1-Owaidy, H. Y. Mohamed. Linearized Oscillation for Nonlinear Systems of Delay Differential Equations. Appl. Math. Comput. 2003, 142(1):17~21
25 J. Dzurina, I. P. Stavroulakis. Oscillation Criteria for Second Order Delay Differential Equations. Appl. Math. Comput. 2003, 140(2-3):445~453
26 J. C. Jiang, X. P. Li. Oscillation of Second Order Nonlinear Neutral Differential Equations. Appl. Math. Comput. 2003, 135(2-3):531~540
27 Y. Sahiner. On Oscillation of Second Order Neutral Type Delay Differential Equations. Appl. Math. Comput. 2004, 150(3):697~706
28 J. H. Shen, X. H. Tang. New Nonoscillation Criteria for Delay Differential Equations. J. Math. Anal. Appl. 2004, 290(1):1~9
29 A. Tiryaki, M. F. Aktas. Oscillation Criteria of a Certain Class of Third Order Nonlinear Delay Differential Equations with Damping. J. Math. Anal. Appl. 2007, 325(1):54~68
30 A. R. Aftabizadeh, J. Wiener and J. M. Xu. Oscillatory and Periodic Properties of Delay Differential Equations with Piecewise Constant Argument. Proc. Amer. Math. Soc. 1987, 99(4):673~679
31 G. Lada. Oscillations of Equations with Piecewise Constant Mixed Arguments. Differential Equations and Applications. Ohio University Press. Athens. 1988:64~69
32 I. Gy o?? ri, G. Ladas. Linearized Oscillations for Equations with Piecewise Constant Arguments. Differential and Integral Equations. 1989(2):123~131
33 J. H. Shen, I. P. Stavroulakis. Oscillatory and Nonoscillatory of Delay Equations with Piecewise Constant Argument. J. Math. Anal. Appl. 2000, 248(2):385~401
34 Z. G. Luo, J. H. Shen. New Results on Oscillation for Delay Differential Equations with Piecewise Constant Arguments. Comput. Math. Appl. 2003, 45(12):1841~1848
35 Y. B. Wang, J. R. Yan. Oscillation of a Differential Equation with Fractional Delay and Piecewise Constant Arguments. Comput. Math. Appl. 2006, 52(6):1099~1106
36 Y. Xu, M. Z. Liu. H-stability of Linearθ-method with General Variable Stepsize for System of Pantograph Equations with Two Delay Terms. Comput. Math. Appl. 2004, 156(3):817~829
37 E. Hair, G. Wanner. Solving Ordinary Differential EquationsⅡ. Stiff and Differential Algebraic Problems. Springer. New York, 1996,
38匡蛟勋.泛函微分方程的数值处理.科学出版社, 1999:82~102
39 I. Gy o??ri, G. Ladas. Oscillation Theory of Delay Equation: With Applications. Clarendon Press. Oxford, 1991:102-111
40胡建伟,汤怀民.微分方程数值方法.科学出版社,1999:187~212
2 V. B. Kolmanovskii, V. R. Nosov. Stability of Functional Differential Equations. Academic Press. New York, 1986:25~163
3 J. Wiener. Generalized Solutions of Functional Differential Equations. World Scientific. Singapore. 1993:1~212
4 E. A. Grove, I. Gy o??ri and G. Ladas. On the Characteristic Equations for Equations with Continuous and Piecewise Constant Delays. Rad. Mat. 1989, 5(2):271~281
5 K. N. Jayasree, S. G. Deo. Variation of Parameters Formula for the Equation of Cooke and Wiener. Proc. Amer. Math. Soc. 1991, 112(1):75~80
6 K. L. Cooke, J. Wiener. Neutral Differential Equation with Piecewise Constant Argument. Boll. Un. Mat. Ital. 1987, 7(1):321~346
7 W. J. Lv, Z. W. Yang and M. Z. Liu. Stability of the Euler-Maclaurin Methods for Neutral Differential Equations with Piecewise Continuous Arguments. Appl. Math. Comput. 2007, 186(2):1480~1487
8 W. J. Lv, Z. W. Yang and M. Z. Liu. Stability of Runge-Kutta Methods for the Alternately Advanced and Retarded Differential Equations with Piecewise Continuous Arguments. Comput. Math. Appl. 2007, 54(3):326~335
9 M. Z. Liu, S. F. Ma and Z. W. Yang. Stability Analysis of Runge-Kutta Methods for Unbounded Retarded Differential Equations with Piecewise Continuous Arguments. Appl. Math. Comput. 2007, 191(1):57~66
10 Z. W. Yang, M. Z. Liu and M. H. Song. Stability of Runge-Kutta Methods in the Numerical Solution of Equation u′( t ) =au(t ) +a0 u([ t ]) +a1u([ t ?1]). Appl. Math. Comput. 2005, 162(1):37~50
11 M. Z. Liu, M. H. Song and Z. W. Yang. Stability of Runge-Kutta Methods in the Numerical Solution of Equation u′( t ) =au(t ) +a0u([ t ]). J. Comput. Appl. Math. 2004, 166(2):361~370
12 M. H. Song, Z. W. Yang and M. Z. Liu. Stability ofθ-Methods for Advanced Differential Equations with Piecewise Continuous Arguments. Comput. Math. Appl. 2005, 49(9-10):1295~1301
13 N. Guglielmi, M. Zennaro. On the Zero-stability of Variable Stepsize Multistep Methods: the Spectral Radius Approach. Numer. Math. 2001, 88(3):445~458
14 K. J. in’t Hout. On the Stability of Adaptations of Runge-Kutta Methods to Systems of Delay Differential Equations. Appl. Num. Math. 1996, 22(1),237-250
15 T. Koto. Stability ofθ-methods for Delay Integro-differential Equations. J. Comput. Appl. Math. 2003, 161(2):393~404
16 K. L. Cooke, J. Wiener. Stability Regions for Linear Equations with Piecewise Continuous Arguments. Comput. Math. Appl. 1986, 12(6):695~701
17吕万金,刘明珠.方程x′( t ) =a x (t ) + a2 x ([t + 2])的线性θ-方法数值稳定性.黑龙江大学自然科学学报. 2006, 23(5):700~702
18 S. M. Shah, J. Wiener. Advanced Differential Equations with Piecewise Constant Argument Deviations. Int. J. Math. Soc. 1983, 6(4):671~703
19 L. Torelli. A Sufficient Condition for GPN-Stability for Delay Differential Equations. Numer. Math. 1991, 59(1):311~320
20 G. Ladas. Y. G. Sficas and I. P. Stavroulakis. Necessary and Sufficient Conditions for Oscillations. AM. Math. mon. 1983, 90(9):637~640
21 I. Kubiaczyk, S. H. Saker. Oscillation and Stability in Nonlinear Delay Differential Equations of Population Dynamics. Math. Comput. Model. 2002, 35(3-4):295~301
22 L. Berezansky, Y. Domshlak and E. Braverman. On Oscillation Properties of Delay Differential Equations with Positive and Negative Coefficients. J. Math. Anal. Appl. 2002, 274(1):81~101
23 I. Kubiaczyk, S. H. Saker and J. Morchalo. New Oscillation Criteria for First Order Nonlinear Neutral Delay Differential Equations. Appl. Math. Comput. 2003, 142(2-3):225~242
24 H. E1-Owaidy, H. Y. Mohamed. Linearized Oscillation for Nonlinear Systems of Delay Differential Equations. Appl. Math. Comput. 2003, 142(1):17~21
25 J. Dzurina, I. P. Stavroulakis. Oscillation Criteria for Second Order Delay Differential Equations. Appl. Math. Comput. 2003, 140(2-3):445~453
26 J. C. Jiang, X. P. Li. Oscillation of Second Order Nonlinear Neutral Differential Equations. Appl. Math. Comput. 2003, 135(2-3):531~540
27 Y. Sahiner. On Oscillation of Second Order Neutral Type Delay Differential Equations. Appl. Math. Comput. 2004, 150(3):697~706
28 J. H. Shen, X. H. Tang. New Nonoscillation Criteria for Delay Differential Equations. J. Math. Anal. Appl. 2004, 290(1):1~9
29 A. Tiryaki, M. F. Aktas. Oscillation Criteria of a Certain Class of Third Order Nonlinear Delay Differential Equations with Damping. J. Math. Anal. Appl. 2007, 325(1):54~68
30 A. R. Aftabizadeh, J. Wiener and J. M. Xu. Oscillatory and Periodic Properties of Delay Differential Equations with Piecewise Constant Argument. Proc. Amer. Math. Soc. 1987, 99(4):673~679
31 G. Lada. Oscillations of Equations with Piecewise Constant Mixed Arguments. Differential Equations and Applications. Ohio University Press. Athens. 1988:64~69
32 I. Gy o?? ri, G. Ladas. Linearized Oscillations for Equations with Piecewise Constant Arguments. Differential and Integral Equations. 1989(2):123~131
33 J. H. Shen, I. P. Stavroulakis. Oscillatory and Nonoscillatory of Delay Equations with Piecewise Constant Argument. J. Math. Anal. Appl. 2000, 248(2):385~401
34 Z. G. Luo, J. H. Shen. New Results on Oscillation for Delay Differential Equations with Piecewise Constant Arguments. Comput. Math. Appl. 2003, 45(12):1841~1848
35 Y. B. Wang, J. R. Yan. Oscillation of a Differential Equation with Fractional Delay and Piecewise Constant Arguments. Comput. Math. Appl. 2006, 52(6):1099~1106
36 Y. Xu, M. Z. Liu. H-stability of Linearθ-method with General Variable Stepsize for System of Pantograph Equations with Two Delay Terms. Comput. Math. Appl. 2004, 156(3):817~829
37 E. Hair, G. Wanner. Solving Ordinary Differential EquationsⅡ. Stiff and Differential Algebraic Problems. Springer. New York, 1996,
38匡蛟勋.泛函微分方程的数值处理.科学出版社, 1999:82~102
39 I. Gy o??ri, G. Ladas. Oscillation Theory of Delay Equation: With Applications. Clarendon Press. Oxford, 1991:102-111
40胡建伟,汤怀民.微分方程数值方法.科学出版社,1999:187~212