HONG Wen-qiang, XU Ji-qing, XU Xi-bin, ZHANG Chun, ZHOU Shi-liang. The Radial Basis Function Approximation Method for Solving Bratu-Type Equations[J]. Applied Mathematics and Mechanics, 2016, 37(6): 617-625. doi: 10.3879/j.issn.1000-0887.2016.06.007
Citation: HONG Wen-qiang, XU Ji-qing, XU Xi-bin, ZHANG Chun, ZHOU Shi-liang. The Radial Basis Function Approximation Method for Solving Bratu-Type Equations[J]. Applied Mathematics and Mechanics, 2016, 37(6): 617-625. doi: 10.3879/j.issn.1000-0887.2016.06.007

The Radial Basis Function Approximation Method for Solving Bratu-Type Equations

doi: 10.3879/j.issn.1000-0887.2016.06.007
Funds:  The National Natural Science Foundation of China,The National Basic Research Program of China (973 Program)
  • Received Date: 2015-12-24
  • Rev Recd Date: 2016-02-15
  • Publish Date: 2016-06-15
  • Based on the powerful approximation capability of the radial basis function for almost all kinds of functions, and with reference to the interpolation method for elasto-plastic mechanics, the radial basis function expression of the interpolation combining displacement, velocity and acceleration was put forward. Then the MATLAB software was used for computer programming to successfully solve the strongly nonlinear Bratu-type equation, with the corresponding relative errors given and discussed. The analysis of several typical examples was conducted, where the present calculated results were compared with some of the existing numerical results as well as the exact solutions. The comparison shows the feasibility and high accuracy of the present method, which makes a new way of solving the strongly nonlinear Bratu-type equations.
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  • [1]
    Abolarin O E. New improved variational homotopy perturbation method for Bratu-type problems[J]. American Journal of Computational Mathematics,2013,3(2): 110-113.
    Syam M I, Hamdan A. An efficient method for solving Bratu equations[J]. Applied Mathematics and Computation,2006,176(2): 704-713.
    Boyd J P. An analytical and numerical study of the two-dimensional Bratu equation[J]. Journal of Scientific Computing,1986,1(2): 183-206.
    Boyd J P. Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one dimensional Bratu equation[J]. Applied Mathematics and Computation,2003,143(2/3): 189-200.
    Jacobson J, Schmitt K. The Liouville-Bratu-Gelfand problem for radial operators[J]. Journal of Differential Equations,2002,184(1): 283-298.
    E·克鲁泽. 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.(Chinese version))
    胡海岩. 应用非线性动力学[M]. 北京: 航空工业出版社, 2000.(HU Hai-yan. Applied Nonlinear Transient Dynamical [M]. Beijing: Aviation Industry Press, 2000.(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, 徐自力, 郑李坤. 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, XU Zi-li, ZHENG Li-kun. A multigrid preconditioned conjugate gradient 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 Numerica 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))
    Aregbesola Y A S. Numerical solution of Bratu problem using the method of weighted residual[J]. Electronic Journal of Southern African Mathematical Sciences,2003,3(1): 652-663.
    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))
    陈文, 傅卓佳, 魏星. 科学与工程计算中的径向基函数方法[M]. 北京: 科学出版社, 2014.(CHEN Wen, FU Zhuo-jia, WEI Xing. The Radial Basis Function Methods in Science and Engineering Mathmatics [M]. Beijing: Science Press, 2014.(in Chinese))
    马利敏. 径向基函数逼近中的若干理论、方法及其应用[D]. 博士学位论文. 上海: 复旦大学, 2009.(MA Li-min. Some theory, methods and application in RBF approaching[D]. PhD Thesis. Shanghai: Fudan University, 2009.(in Chinese))
    Lin J, Chen W, Sze K Y. A new radial basis function for Helmholtz problems[J]. Engineering Analysis With Boundary Elements,2012,36(12): 1923-1930.
    Fu Z J, Chen W, Ling L. Method of approximate particular solutions for constant- and variable-order fractional diffusion models[J]. Engineering Analysis With Boundary Elements,2015,57: 37-46.
    徐绩青, 李正良, 吴林键. 基于径向基函数逼近的结构动力响应计算方法[J]. 应用数学和力学, 2014,35(5): 533-541.(XU Ji-qing,LI Zheng-liang,WU Lin-jian. A calculation method for structural dynamic responses based on the approximation theory of radial basis function[J]. Applied Mathematics and Mechanics,2014,35(5): 533-541.(in Chinese))
    李岩汀, 许锡宾, 周世良, 徐绩青. 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, XU Ji-qing. 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))
    Abbasbandy S, Hashemi M S, Liu C-S. The Lie-group shooting method for solving the Bratu equation[J]. Communications in Nonlinear Science and Numerical Simulation,2011,16(11): 4238-4249.
    Deeba E, Khuri S A, XIE Shi-shen. An algorithm for solving boundary value problems[J]. Journal of Computational Physics,2000,159(2): 125-138.
    Khuri S A. A new approach to Bratu’s problem[J]. Applied Mathematics and Computation,2004,147(1): 131-136.
    Caglar H, Caglar N, ?zer M, Valaristos A, Anagnostopoulos A N. B-spline method for solving Bratu’s problem[J]. International Journal of Computer Mathematics,2010,87(8): 1885-1891.
    Jalilian R. Non-polynomial spline method for solving Bratu’s problem[J]. Computer Physics Communications,2010,181(11): 1862-1872.
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