再生核空间中对非自治系统的求解
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摘要
微分方程起源于各种应用学科中,例如核物理、气体动力学、流体力学、边界层理论、非线性光学等。非自治系统的研究是微分方程的一个重要内容,在实际应用中,许多物理问题都可以转化为求解非自治系统(微分方程组)的问题,通过设置人工势场和人工边界条件的办法去控制微观粒子的量子运动,非自治系统的哈密顿量或边界条件通过某些参数依赖于时间,这些随时间变化的参数体现了人类对系统的控制或环境对系统的影响,因此对非自治系统算法及理论的研究对推动量子系统的发展非常重要。能否得到适当精度和可靠的数值解在很大程度上依赖所用的数值方法。精确度高、稳定性强、收敛性好、计算量少的算法显得尤为重要。本文在再生核空间中给出了非自治系统的求解方法。对于线性的微分方程组,主要利用再生核函数构造出再生核空间的完全正交系,将非自治系统的解以级数形式精确的给出,并通过对精确解的级数进行截断,从而得到其近似解。方法的创新之处在于证明了近似解收敛到精确解。对于非线性微分方程组,构造了一个迭代序列,证明了迭代序列是有界的,利用迭代序列构造近似解的表达形式。方法的特点在于对任意给定的初值函数,近似解收敛到方程的精确解。本文给出了一些数值算例来验证我们的方法的精度,数值结果表明了本文的所提方法的可行性和有效性。
Differential equations arise from many applied fields, such as nuclear physics, pneumatic dynamics, fluid mechanics, boundary layer theory, nonlinear optics and so on. Non-autonomous systems are a class of important problems of the differential equations. In practical applications, a lot of physical problems can be reduced to solving the problem of nonautonomous system. One can control microscopic particles of quantum movement by setting artificially potential field and boundary conditions. The Hamilton quantum or boundary conditions of nonautonomous system depend on time through certain parameters. These parameters which change with time reflect that the human controls the system or the system is affected by the environmental impact. Thus the study of theories and algorithms for nonautonomous systems are very important for the development of quantum systems. It is important to obtain the algorithms which possess the excellent properties such as high accuracy, strong stability, good convergence and little labor of calculation. In this paper, the method for solving the non-autonomous system is proposed in reproducing kernel space. For a linear system of ordinary differential equations, using the reproducing kernel function, the complete normal orthogonal system of the reproducing kernel space is constructed. The exact solution of the system is represented in the form of series. By truncating the series, the approximate solution is obtained. The advantage of the approach is that the approximate solution uniformly converges to the exact solution. For a nonlinear system of ordinary differential equations, we construct an iterative sequence and prove that the sequence is bounded. Using the sequence, the approximate solution is obtained. For an arbitrary value function, the approximate solution converges uniformly to exact solution. We give some numerical examples to validate the present method. And the numerical results demonstrate the feasibility and validity of our method.
Differential equations arise from many applied fields, such as nuclear physics, pneumatic dynamics, fluid mechanics, boundary layer theory, nonlinear optics and so on. Non-autonomous systems are a class of important problems of the differential equations. In practical applications, a lot of physical problems can be reduced to solving the problem of nonautonomous system. One can control microscopic particles of quantum movement by setting artificially potential field and boundary conditions. The Hamilton quantum or boundary conditions of nonautonomous system depend on time through certain parameters. These parameters which change with time reflect that the human controls the system or the system is affected by the environmental impact. Thus the study of theories and algorithms for nonautonomous systems are very important for the development of quantum systems. It is important to obtain the algorithms which possess the excellent properties such as high accuracy, strong stability, good convergence and little labor of calculation. In this paper, the method for solving the non-autonomous system is proposed in reproducing kernel space. For a linear system of ordinary differential equations, using the reproducing kernel function, the complete normal orthogonal system of the reproducing kernel space is constructed. The exact solution of the system is represented in the form of series. By truncating the series, the approximate solution is obtained. The advantage of the approach is that the approximate solution uniformly converges to the exact solution. For a nonlinear system of ordinary differential equations, we construct an iterative sequence and prove that the sequence is bounded. Using the sequence, the approximate solution is obtained. For an arbitrary value function, the approximate solution converges uniformly to exact solution. We give some numerical examples to validate the present method. And the numerical results demonstrate the feasibility and validity of our method.
引文
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4王顺金,李福利,左维,揭泉林,韦联福.量子系统的动力学对称性研究.量子光学学报,2001,6(1):1~17
5 Per Grove Thomsen, Zahari Zlatev. Development of A Data Assimilation Algorithm. Comput. Math. Appli..2008,55(10): 2381~2393
6 J. Wang, H. Zhang. Algebraic Dynamics Solutions and Algebraic Dynamics Algorithm for Nonlinear Partial Differential Evolution Equations of Dynamical Systems. Sci. G-Phys. Mecha. Astr.. 2008, 51(6):577~590
7全景才,朱燊权,丘衡.求非线性动态网络稳定周期解的一种数值算法.华南理工大学报.1995,23(8):68~75
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9 X. Wu. Periodic Solutions for Nonautonomous Second-order Systems with Bounded Nonlinearity. J. Math. Anal. Appl. .1999,230 (1):135~141
10唐美兰,刘心歌,刘心笔.一类差分方程周期解的存在性.山东理工大学学报.2005,19(1):56~60
11罗娜,孙树林.有界约束非线性方程组的仿射内点法.河南科技大学理学院.2008,10:1~8
12 F.Facchinei, S.Lucidi, L.Palagi. A Truncated Newton Algorithm for Large Scale Box Constrained Optimization . SIAM. JOptim.. 2002, 12 :1100~1125
13 C.Kanzow, H.D.Qi,AQP. Free Constrained Newton-type Method for Variational Inequality Problems. Mathe.Prog..1999,85:81~106
14刘星果.解非线性病态方程组的一种修正Newton法及其应用.湖南大学硕士论文.2003:15~35
15 Qaisi M I, Abu-Hilal. Analysis of A Forced Strongly Nonlinear Two-degree-of-freedom System by Means of The Power-series Method. Shock. Vibr. Dig.. 2001,33:78~101
16裘春航,吕和祥.非线性动力学方程的一种级数解.计算力学学报. 2001,18(1):1~7
17 Oz h R. Non-linear Vibrations and Stability Analysis of Tensioned Pipes Conveying Fluid with Variable Velocity. Int. J. Non-linear. Mech.. 2001, 36 :1031~1039
18 Li Li. Energy Method for Approximate Periodic Solution of Strongly Nonlinear Nonautonomous Systems. Nonlinear Dynamics. 1999, 19:101~120
19 H. Chen, K. Chang. A Modified Lindstedt-Poincare Method for A Strongly Nonlinear System with Quadratic and Cubic Nonlinearities. Shock. Vibr.. 1996,3 (4):15~70
20唐驾时.求强非线性系统次谐共振解的MLP方法.应用数学和力学.2000,10:110~120
21周一峰.求强非线性动力系统周期解的MH法和EMH法.中南大学学报. 2004,01(30):1005~9792
22裘春航,吕和祥.在哈密顿体系下分析非线性动力学问题.计算力学学报.2000,17(2):169~172
23周一峰,唐进元,何旭辉.二阶强非线性非自治系统周期解的能量迭代法.振动与冲击.2006,25(3):193~197
24 H.B. Thompson, C. Tisdell. Systems of Difference Equations Associated with Boundary Value Problems for Second Order Systems of Ordinary Differential Equations. J. Math. Anal. Appl.. 2000,248:333~347
25 H. B. Thompson, C. Tisdell. Boundary Value Problems for Systems of Difference Equations Associated with Systems of Second-order Ordinary Differential Equations. Appl. Math. Lett.. 2002,15(6):761~766
26 S. Li, P. Liu, M. Jia, L. Li, F. Li. Existence of Three Positive Solutions in Boundary Value Problems of A Class of Second Order Ordinary Differential Systems. J. Shu. Sci. Tech.. 2007,1:301~312
27 Y. Liu, X. Liu, M. Jia. Multiplicity Results for Second Order m Point Boundary Value Problem. J. Math. Anal.Appl..2006, 324 :532~542
28 H.B. Thompson. Systems of Differential Equations with Fully Nonlinear Boundary Conditions. Austd. Math. Sot.. 1997(56):197~208
29 X. Cheng, C. Zhong. Existence of Positive Solutions for A Second Order Ordinary Differential System. J. Math. Anal. Appl.. 2005(312):14~23
30 C.O. Alves, D.G. De Figueiredo. Nonvariational Elliptic Systems. Discrete Contin. Dyn. Syst.. 2002(8 ):289~302
31 X. Cheng, C. Zhong. Existence of Positive Solutions for A Second-orderOrdinary Differential System. J. Math. Anal. Appl.. 2005,312:1~23
32 D.R. Dunninger, H. Wang. Existence and Multiplicity of Positive Solutions for Elliptic Systems. Nonlinear Anal.. 1997,29:1051~1060
33 D.R. Dunninger, H. Wang. Multiplicity of Positive Radial Solutions for An Elliptic System on An Annulus. Nonlinear Anal.. 2000,42: 803~811
34 Y.H. Lee. Multiplicity of Positive Radial Solutions for Multiparameter Semilinear Elliptic Systems on An Annulus. J. Differential Equations. 2001,174:420~441
35 R. Ma. Existence of Positive Radial Solutions for Elliptic Systems. J. Math. Anal. Appl.. 1996(201):375~386
36 Joa? Marcos doó, Sebastián Lorca, Pedro Ubilla. Local Superlinearity for Elliptic Systems Involving Parameters. J. Differential Equations. 2005, 211: 312~320
37 J.M. doó, S. Lorca, J. Sánchez, P. Ubilla. Positive Solutions for A Class of Multiparameter Ordinary Elliptic Systems. J. Math. Anal. Appl.. 2007,332(2):1249~1266
38 Y. Liu. Multiple Positive Solutions of Nonlinear Singular Boundary Value Problem for Fourth-order Equations. Appl. Math. Lett.. 2004,17: 747~757
39 R. Ma, C. Tisdell. Positive Solutions of Singular Sublinear Fourth-order Boundary Value Problems. Appl. Anal.. 2005,84:1199~1220
40 H.B. Thompson, C. Tisdell. The Nonexistence of Spurious Solutions to Discrete Two-point Boundary Value Problems. Appl. Math. Lett.. 2003,16 (1): 79~84
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