留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

基于MQ拟插值函数逼近的非线性动力系统数值求解

杜珊 李风军

杜珊, 李风军. 基于MQ拟插值函数逼近的非线性动力系统数值求解[J]. 应用数学和力学, 2017, 38(8): 943-955. doi: 10.21656/1000-0887.370368
引用本文: 杜珊, 李风军. 基于MQ拟插值函数逼近的非线性动力系统数值求解[J]. 应用数学和力学, 2017, 38(8): 943-955. doi: 10.21656/1000-0887.370368
DU Shan, LI Feng-jun. A Numerical Approximation Method for Solutions to Nonlinear Dynamic Systems Based on Multiquadric Quasi-Interpolation Functions[J]. Applied Mathematics and Mechanics, 2017, 38(8): 943-955. doi: 10.21656/1000-0887.370368
Citation: DU Shan, LI Feng-jun. A Numerical Approximation Method for Solutions to Nonlinear Dynamic Systems Based on Multiquadric Quasi-Interpolation Functions[J]. Applied Mathematics and Mechanics, 2017, 38(8): 943-955. doi: 10.21656/1000-0887.370368

基于MQ拟插值函数逼近的非线性动力系统数值求解

doi: 10.21656/1000-0887.370368
基金项目: 国家自然科学基金(11261024;61662060)
详细信息
  • 中图分类号: O29;O302

A Numerical Approximation Method for Solutions to Nonlinear Dynamic Systems Based on Multiquadric Quasi-Interpolation Functions

Funds: The National Natural Science Foundation of China(11261024;61662060)
  • 摘要: 借助多重二次曲面(multi quadrics,MQ)拟插值函数具有较好精确性和稳定性的优势,研究了基于MQ拟插值函数和4阶Runge-Kutta法相结合的方法,构造了求解带有初值问题的非线性动力系统的数值解法,分析了该方法与已有主要方法的优缺点,并给出了相应的数值算例、误差估计.结果表明该方法计算量小、能很好地逼近非线性动力系统的解析解.
  • [1] DAI Hong-hua, YUE Xiao-kui, YUAN Jiang-ping, et al. Half-order optimally scaled Fourier expansion method for solving nonlinear dynamical system[J]. International Journal of Non-Linear Mechanics,2016,87: 21-29.
    [2] LIU Chein-shan. A novel Lie-group theory and complexity of nonlinear dynamical systems[J]. Communications in Nonlinear Science and Numerical Simulation,2015,20(1): 39-58.
    [3] LIU Yan-jun, TONG Shao-cheng. Adaptive fuzzy control for a class of unknown nonlinear dynamical systems[J].Fuzzy Sets and Systems,2015,263(15): 49-70.
    [4] Zeng Y, Zhu W Q. Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations[J]. International Journal of Non-Linear Mechanics,2010,45(5): 572-586.
    [5] Chen Y M, Liu J K. A precise calculation of bifurcation points for periodic solution in nonlinear dynamical systems[J].Applied Mathematics and Computation,2016,273: 1190-1195.
    [6] E·克鲁译. 非线性动力学系统的数值研究[M]. 凌复华, 译. 上海:上海交通大学出版社, 1989.(Kreuzer E. Numerische Untersuchung Nichtlinearer Dynamischer Systeme [M]. LING Fu-hua, transl. Shanghai: Shanghai Jiao Tong University Press, 1989.(in Chinese))
    [7] 刘向军, 石磊, 徐旭常. 稠密气固两相流欧拉-拉格朗日法的研究现状[J]. 计算力学学报, 2007,24(2): 166-172.(LIU Xiang-jun, SHI Lei, XU Xu-chang. Activities of dense particle-gas two-phase flow modeling in Eulerian-Lagrangian approach[J]. Chinese Journal of Computational Mechanics,2007,24(2): 166-172.(in Chinese))
    [8] 刘石, 陈德祥, 冯永新, 等. 等几何分析的多重网格共轭梯度法[J]. 应用数学和力学, 2014,35(6): 630-639.(LIU Shi, CHEN De-xiang, FENG Yong-xin, et al. A multigrid preconditioned conjugate method for isogeometric analysis[J]. Applied Mathematics and Mechanics,2014,35(6): 630-639.(in Chinese))
    [9] 陈全发, 肖爱国. Runge-Kutta-Nystrm方法的若干新性质[J]. 计算数学, 2008,30(2): 201-212.(CHEN Quan-fa, XIAO Ai-guo. Some new properties of Runge-Kutta-Nystrm methods[J]. Mathematic Numeric Sinica,2008,30(2): 201-212.(in Chinese))
    [10] 樊文欣, 杨桂通, 岳文忠. 基于ADAMS的发动机动力学通用分析模型[J]. 应用基础与工程科学学报, 2009,17(S1): 172-178.(FAN Wen-xin, YANG Gui-tong, YUE Wen-zhong. The dynamic universal analysis model of engine based on ADAMS[J]. Journal of Basic Science and Engineering,2009,17(S1): 172-178.(in Chinese))
    [11] Hardy R L. Multiquadric equations of topography and other irregular surfaces[J]. Journal of Geophysical Research,1971,76(8): 1905-1915.
    [12] Beatson R K, Dyn N. Multi-quadric B-splines[J]. Journal of Approximation Theory,1986,87(1): 1-24.
    [13] Hon Y C, Mao X Z. An efficient numerical scheme for Burgers’ equation[J]. Applied Mathematics and Computation,1998,95: 37-50.
    [14] WU Zong-min, Schaback R. Shape preserving properties and convergence of univariate multiquadric quasi-interpolation[J]. Acta Mathematicae Applicatae Sinica,1994,10(4): 441-446.
    [15] MA Li-min, WU Zong-min. Approximation to the k-th derivatives by multiquadric quasi-interpolation method[J]. Journal of Computational and Applied Mathematics,2009,231(2): 925-932.
    [16] MA Li-min, WU Zong-min. Stability of multiquadric quasi-interpolation to approximate high order derivatives[J]. Science China Mathematics,2010,53(4): 985-992.
    [17] Hardy R L. Theory and applications of the multiquadric-biharmonic method: 20 years of discovery 1968-1988[J]. Computers & Mathematics With Applications,1990,19(8/9): 163-208.
    [18] Buhmann M D.Radial Basis Functions: Theory and Implementations [M]. Cambridge: Cambridge University Press, 2003.
    [19] WU Hui-yuan, DUAN Yong. Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis-Procesi equation[J]. Applied Mathematics and Computation,2016,274: 83-92.
    [20] GAO Wen-wu, WU Zong-min. Solving time-dependent differential equations by multiquadric trigonometric quasi-interpolation[J]. Applied Mathematics and Computation,2015,253: 377-386.
    [21] GAO Feng, CHI Chun-mei. Numerical solution of nonlinear Burger’ equations using high accuracy multi-quadric quasi-interpolation[J]. Applied Mathematics and Computation,2014,229: 414-421.
    [22] WU Zong-min, ZHANG Sheng-liang. Conservative multiquadric quasi-interpolation method for Hamiltonian wave equations[J]. Engineering Analysis With Boundary Elements,2013,37(7/8): 1052-1058.
    [23] Franke R. Scattered data interpolation: text of some methods[J]. Mathematics of Computation,1982,38: 181-200.
    [24] Madych W R, Nelson S A. Error bounds for multiquadric interpolation[J]. Approximation Theory,1991, 12: 413-416.
    [25] Buhmann M D. Multivariate interpolation in odd-dimensional Euclidean space using multiquadrics[J]. Constructive Approximation,1990,6(1): 21-34.
    [26] Beatson R, Powell M J D. Univariate multiquadric approximation: quasi-interpolation to scattered data[J]. Constructive Approximation,1992,8(3): 275-288.
    [27] Power M J D. Univariate multiquadric approximation: reproduction of linear polynomials multivariate approximation and interpolation[M]//Haumann W, Jetter K. Multivariate Approximation and Interpolation . Birkhuser Basel, 1990: 227-240.
    [28] 李庆阳, 王能超, 易大义. 数值分析[M]. 北京: 清华大学出版社, 2008.(LI Qing-yang, WANG Neng-chao, YI Da-yi. Numerical Analysis [M]. Beijing: Tsinghua University Press, 2008.(in Chinese))
    [29] 李鹏柱, 李风军, 李星, 等. 基于三次样条插值函数的非线性动力系统数值求解[J]. 应用数学和力学, 2015,36(8): 887-896.(LI Peng-zhu, LI Feng-jun, LI Xing, et al. A numerical method for the solution to nonlinear dynamic systems based on cubic spline interpolation functions[J]. Applied Mathematics and Mechanics,2015,36(8): 887-896.(in Chinese))
    [30] 李岩汀, 许锡宾, 周世良, 等. 基于径向基函数逼近的非线性动力系统数值求解[J]. 应用数学和力学, 2016,37(3): 311-318.(LI Yan-ting, XU Xi-bin, ZHOU Shi-liang, et al. A numerical approximation method for nonlinear dynamic systems based on radial basis functions[J]. Applied Mathematics and Mechanics,2016,37(3): 311-318.(in Chinese))
    [31] 吴宗敏. 径向基函数、散乱数据拟合与无网格偏微分方程数值解[J]. 工程数学学报, 2002,19(2): 1-12.(WU Zong-min. Radial basis function scattered data interpolation and the meshless method of numerical solution of PDEs[J]. Journal of Engineering Mathematics,2002,19(2): 1-12.(in Chinese))
    [32] 刘卫国. MATLAB程序设计与应用[M]. 第5版. 北京: 高等教育出版社, 2008.(LIU Wei-guo. MATLAB Program Design and Application [M]. 5th ed. Beijing: Higher Education Press, 2008.(in Chinese))
  • 加载中
计量
  • 文章访问数:  690
  • HTML全文浏览量:  72
  • PDF下载量:  670
  • 被引次数: 0
出版历程
  • 收稿日期:  2016-11-29
  • 修回日期:  2017-01-13
  • 刊出日期:  2017-08-15

目录

    /

    返回文章
    返回