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

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

doi: 10.21656/1000-0887.370368
Funds:  The National Natural Science Foundation of China(11261024;61662060)
  • Received Date: 2016-11-29
  • Rev Recd Date: 2017-01-13
  • Publish Date: 2017-08-15
  • The multiquadric quasi-interpolation function has advantages of high accuracy and good stability. A new numerical method for resolving the initial value problems of nonlinear dynamic systems was proposed via combination of the multiquadric quasi-interpolation function and the 4th-order Runge-Kutta method. The advantages and disadvantages were analyzed between this new method and the existing numerical methods for nonlinear dynamic systems, according to the numerical example and error estimation. The results show that the proposed method needs less computation cost and enables fine approximation to the analytical solutions to nonlinear dynamic systems.
  • loading
  • [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.
    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.
    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.
    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.
    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.
    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))
    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))
    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))
    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))
    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))
    Hardy R L. Multiquadric equations of topography and other irregular surfaces[J]. Journal of Geophysical Research,1971,76(8): 1905-1915.
    Beatson R K, Dyn N. Multi-quadric B-splines[J]. Journal of Approximation Theory,1986,87(1): 1-24.
    Hon Y C, Mao X Z. An efficient numerical scheme for Burgers’ equation[J]. Applied Mathematics and Computation,1998,95: 37-50.
    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.
    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.
    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.
    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.
    Buhmann M D.Radial Basis Functions: Theory and Implementations [M]. Cambridge: Cambridge University Press, 2003.
    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.
    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.
    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.
    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.
    Franke R. Scattered data interpolation: text of some methods[J]. Mathematics of Computation,1982,38: 181-200.
    Madych W R, Nelson S A. Error bounds for multiquadric interpolation[J]. Approximation Theory,1991, 12: 413-416.
    Buhmann M D. Multivariate interpolation in odd-dimensional Euclidean space using multiquadrics[J]. Constructive Approximation,1990,6(1): 21-34.
    Beatson R, Powell M J D. Univariate multiquadric approximation: quasi-interpolation to scattered data[J]. Constructive Approximation,1992,8(3): 275-288.
    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.
    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))
    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))
    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))
    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))
    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


    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (1180) PDF downloads(679) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint