微分方程数值计算及稳定性研究
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摘要
本文主要研究的是关于关于微分代数系统解的迭代序列的收敛性、波形松弛迭代法收敛性的新的方法、具有时滞扰动的非自治中立型FDE的3/2-稳定性。
在第—章中,得到了微分代数系统波形松弛迭代法的收敛性的判别准则。
在第二章中,给出了判断方程波形松弛迭代法收敛性的新的方法。
在第三章中,我们讨论了具有时滞扰动的非自治中立型FDE的3/2-稳定性,得到了零解一致稳定和渐近稳定的充分条件。
In this paper, we mainly discuss the convergence condition of iterative sequence of the solution , a new method for the convergence on wave relaxation, asymptotic stability for a class of perturbing nonautonomous neutral differential equations.In chapter 1, we obtain the convergence condition of iterative sequence of the solutionofIn chapter 2, we obtain a new method for the convergence on wave relaxationIn chapter 3, we investigate the 3/2-stability for perturbibg nonautonomous neutral differential equations with two different delay and obtain sufficient conditions for the zero solution of this equation to be uniformly stable as well as asymptotically stable . The criteria are of simple forms , easily checked , and applicable.
在第—章中,得到了微分代数系统波形松弛迭代法的收敛性的判别准则。
在第二章中,给出了判断方程波形松弛迭代法收敛性的新的方法。
在第三章中,我们讨论了具有时滞扰动的非自治中立型FDE的3/2-稳定性,得到了零解一致稳定和渐近稳定的充分条件。
In this paper, we mainly discuss the convergence condition of iterative sequence of the solution , a new method for the convergence on wave relaxation, asymptotic stability for a class of perturbing nonautonomous neutral differential equations.In chapter 1, we obtain the convergence condition of iterative sequence of the solutionofIn chapter 2, we obtain a new method for the convergence on wave relaxationIn chapter 3, we investigate the 3/2-stability for perturbibg nonautonomous neutral differential equations with two different delay and obtain sufficient conditions for the zero solution of this equation to be uniformly stable as well as asymptotically stable . The criteria are of simple forms , easily checked , and applicable.
引文
[1] J. K. White and A. Sangiovanni-Vincentelli, Relaxation Techniques for the Simulation of VLSI Circuits, Kulwer Academic Publishers, Boston, 1986.
[2] U·Miekkala and O. Nevanlinna, Iterative solution of systems of linear diferential equations. Actn Numerica, (1996), 259-307.
[3] Y. L. Jiang, W . S. Lul(?), and O. Wing, Convergence-theoretics of classical and Krylov waveform relaxation methods for diferential-algebraic equations, IEICE Transactions on Fundamentals of ec ￡ mnics, Communications, and Computer Sciences, E80-A. 10(1997), 1961-1972.
[4] Y. L. Jiang and O. Wing, Monotone waveform relaxation for systems of nonlinear diferentialalgebraic equations, SIAM Journal on Numerical Analysis, 38: 1(2000), 170-185.
[5] Y. L. Jiang, R. M . M. Chen, and O. Wing, Improving convergence performance of relaxation-based transient analysis by matrix splitting in circuit simulation, IEEE Transactions on Circuits and Systems. Part 48: 6(2001), 769-780.
[6] Z. L. Huang, R. M. M. Chen, and Y. L. Jiang, A parallel decoupling technique to accelerate convergence of relaxation solutions of integral-diferential-algebraic equations, Journal of Interconnection Networks, 2: 3(2001), 295-304.
[7] M. R. Crisci, E. Rnsso, and A. Vecchio, Time point relaxation methods for Volterra integrodiferential equations, Computers Math. Applic., 36: 9(1998), 59-70.
[8] M. R. Crisci, N. Ferraro, and E. Russo, Convergence results for continuous. time waveform methods for Volterra integral equations, Journal of Computational and Applied Mathematics, 71: 1 (1996), 33-45.
[9] K. R. Schneider. A remark on the wave-form relaxation method. International J. Circuit Theory Appl., 19(1991), 101～104.
[10] Z. Jackiewicz and M. Kwapisz, Convergence of waveform relaxation methods for differentialalgebraic equations, SIAM J. Numer Anal, 33: 6I1996、. 2303-2317.
[11] T. Kato, Perturbation Theoryfor Linear Operators, Springer-Yerlag, Berlin, 1984.
[12] J. Janssen and S. Vandewalle, Multigrid waveform relaxation on spatial finite element meshes: The continuous. time case, SIAM J. Ⅳ umer. Anal., 33: 2 f1996), 456-474.
[13] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.
[14] Y. saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, New York 1996.
[15] Z. Bartoszewski and M. Kwapisz, On error estimates for waveform relaxation methods for delay, diferential-equations, SIAM J. Numer. Anal. 382(2000), 639-659. J . Sand and K. Burrage, A Jacobi waveform relaxation method for ODEs, SIAM J. Sci. Comput., 20, 2(1999), 534-552
[16] Yorke,J.A.,A symptotic stability for one dimensional differential-delay equations[J],J.Diff.Eqs.,7(1970), 189-202.
[17] Yoneyama,T.,On the 3/2 stability theorem for one-dimensional delay-differential equations[J], J.math.Anal.Appl.,125(1987), 161-173.
[18] Yoneyama,T.Sugie,T.,Perturbing uniformly stable nonlinear scalar delay-differentialequations[J], Nonlinear Analysis,12:3(1988),303-311.
[19] 沈根成,高国柱,李立蜂,具有扰动和无界时滞的一维泛函微分方程的渐近稳定性[J],数学物理通报,19:5(1999),493-500.
[20] Yu jianshe, Asymptotic stability for a class of non-autonomous neutral differential equations[J], Chin. Ann. of Math., 18B:4(1997), 449-456.
[21] 叶海平.高国柱.具有扰动的非自治中立型泛函微分方程的渐近稳定性[J],数学年刊,23A:3(2002),389-394.
[2] U·Miekkala and O. Nevanlinna, Iterative solution of systems of linear diferential equations. Actn Numerica, (1996), 259-307.
[3] Y. L. Jiang, W . S. Lul(?), and O. Wing, Convergence-theoretics of classical and Krylov waveform relaxation methods for diferential-algebraic equations, IEICE Transactions on Fundamentals of ec ￡ mnics, Communications, and Computer Sciences, E80-A. 10(1997), 1961-1972.
[4] Y. L. Jiang and O. Wing, Monotone waveform relaxation for systems of nonlinear diferentialalgebraic equations, SIAM Journal on Numerical Analysis, 38: 1(2000), 170-185.
[5] Y. L. Jiang, R. M . M. Chen, and O. Wing, Improving convergence performance of relaxation-based transient analysis by matrix splitting in circuit simulation, IEEE Transactions on Circuits and Systems. Part 48: 6(2001), 769-780.
[6] Z. L. Huang, R. M. M. Chen, and Y. L. Jiang, A parallel decoupling technique to accelerate convergence of relaxation solutions of integral-diferential-algebraic equations, Journal of Interconnection Networks, 2: 3(2001), 295-304.
[7] M. R. Crisci, E. Rnsso, and A. Vecchio, Time point relaxation methods for Volterra integrodiferential equations, Computers Math. Applic., 36: 9(1998), 59-70.
[8] M. R. Crisci, N. Ferraro, and E. Russo, Convergence results for continuous. time waveform methods for Volterra integral equations, Journal of Computational and Applied Mathematics, 71: 1 (1996), 33-45.
[9] K. R. Schneider. A remark on the wave-form relaxation method. International J. Circuit Theory Appl., 19(1991), 101～104.
[10] Z. Jackiewicz and M. Kwapisz, Convergence of waveform relaxation methods for differentialalgebraic equations, SIAM J. Numer Anal, 33: 6I1996、. 2303-2317.
[11] T. Kato, Perturbation Theoryfor Linear Operators, Springer-Yerlag, Berlin, 1984.
[12] J. Janssen and S. Vandewalle, Multigrid waveform relaxation on spatial finite element meshes: The continuous. time case, SIAM J. Ⅳ umer. Anal., 33: 2 f1996), 456-474.
[13] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.
[14] Y. saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, New York 1996.
[15] Z. Bartoszewski and M. Kwapisz, On error estimates for waveform relaxation methods for delay, diferential-equations, SIAM J. Numer. Anal. 382(2000), 639-659. J . Sand and K. Burrage, A Jacobi waveform relaxation method for ODEs, SIAM J. Sci. Comput., 20, 2(1999), 534-552
[16] Yorke,J.A.,A symptotic stability for one dimensional differential-delay equations[J],J.Diff.Eqs.,7(1970), 189-202.
[17] Yoneyama,T.,On the 3/2 stability theorem for one-dimensional delay-differential equations[J], J.math.Anal.Appl.,125(1987), 161-173.
[18] Yoneyama,T.Sugie,T.,Perturbing uniformly stable nonlinear scalar delay-differentialequations[J], Nonlinear Analysis,12:3(1988),303-311.
[19] 沈根成,高国柱,李立蜂,具有扰动和无界时滞的一维泛函微分方程的渐近稳定性[J],数学物理通报,19:5(1999),493-500.
[20] Yu jianshe, Asymptotic stability for a class of non-autonomous neutral differential equations[J], Chin. Ann. of Math., 18B:4(1997), 449-456.
[21] 叶海平.高国柱.具有扰动的非自治中立型泛函微分方程的渐近稳定性[J],数学年刊,23A:3(2002),389-394.