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一类典型二阶非线性微分方程的近似解析解研究

楼智美 王元斌 王鹏

楼智美, 王元斌, 王鹏. 一类典型二阶非线性微分方程的近似解析解研究[J]. 华东师范大学学报(自然科学版), 2018, (4): 129-137. doi: 10.3969/j.issn.1000-5641.2018.04.013
引用本文: 楼智美, 王元斌, 王鹏. 一类典型二阶非线性微分方程的近似解析解研究[J]. 华东师范大学学报(自然科学版), 2018, (4): 129-137. doi: 10.3969/j.issn.1000-5641.2018.04.013
LOU Zhi-mei, WANG Yuan-bin, WANG Peng. A study of approximate analytical solutions of a kind of typical second-order nonlinear different equation[J]. Journal of East China Normal University (Natural Sciences), 2018, (4): 129-137. doi: 10.3969/j.issn.1000-5641.2018.04.013
Citation: LOU Zhi-mei, WANG Yuan-bin, WANG Peng. A study of approximate analytical solutions of a kind of typical second-order nonlinear different equation[J]. Journal of East China Normal University (Natural Sciences), 2018, (4): 129-137. doi: 10.3969/j.issn.1000-5641.2018.04.013

一类典型二阶非线性微分方程的近似解析解研究

doi: 10.3969/j.issn.1000-5641.2018.04.013
基金项目: 

国家自然科学基金 11472177

国家自然科学基金 11772141

详细信息
    作者简介:

    楼智美, 女, 教授, 主要从事分析力学研究.E-mail:louzhimei@usx.edu.cn

  • 中图分类号: O316

A study of approximate analytical solutions of a kind of typical second-order nonlinear different equation

  • 摘要: 在非惯性转动参照系中研究力学体系的运动,常常会出现一类分子分母都含非线性项的二阶非线性微分方程,很难求得其近似解.用Adomian分解法研究了这类典型二阶非线性微分方程的近似解,在给定的初始条件和参数下得到了近似解的解析表达式,并作出了近似解析解的解曲线;与直接用Mathematica软件得到的数值解曲线和用同伦渐近法得到的近似解析解曲线进行了比较,结果表明,在第一个1/4周期时间内,用Adomian分解法得到的近似解解曲线与直接用Mathematica软件得到的数值解曲线十分吻合,并且其误差比用同伦渐近法得到的解曲线更小.
  • 图  1  匀速转动的抛物线形金属丝

    Fig.  1  A parabolic wire rotating at a constant speed

    图  2  根据式(22)作出的曲线(红色曲线), 根据式(23)作出的曲线(蓝色曲线), 数值解曲线(黑色曲线)

    Fig.  2  The red curve is constructed according to Eq.(22), the blue curve is constructed according to Eq.(23), and the black curve is constructed according to the numerical solution

    图  3  根据式(20)作出的曲线(蓝色曲线), 根据式(21)作出的曲线(绿色曲线), 根据式(22)作出的曲线(红色曲线), 数值解曲线(黑色曲线)

    Fig.  3  The blue curve is constructed according to Eq.(20), the green curve is constructed according to Eq.(21), the red curve is constructed according to Eq.(22), and the black curve is made according to the numerical solution

    图  4  根据式(24)作出的曲线(红色曲线), 数值解曲线(黑色曲线)

    Fig.  4  The red curve is constructed according to Eq.(24) and the black curve is constructed according to the numerical solution

  • [1] 周衍柏.理论力学[M].3版.北京:高等教育出版社, 2009:269.
    [2] MARINCA V, HERISANU N. Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic[J]. Journal of Sound and Vibration. 2010, 329:1450-1459. doi:  10.1016/j.jsv.2009.11.005
    [3] 方锦清, 姚伟光.逆算符方法求解非线性动力学方程及其一些应用实例[J].物理学报, 1993, 42(9):1375-1384. doi:  10.7498/aps.42.1375
    [4] BIAZAR J, SHAFIOF S M. A simple algorithm for calculating Adomian polynomials[J]. Int J Contemp Math Sci, 2007, 20(2):975-982. http://www.ams.org/mathscinet-getitem?mr=2371329
    [5] 段俊生. Adomian多项式的计算及其在整数阶和分数阶非线性微分方程中的应用[J].应用数学与计算数学学报, 2015, 29(2):187-210. http://www.cnki.com.cn/Article/CJFDTOTAL-YONG201502008.htm
    [6] ZHU Y G, CHANG Q S, WU S C. A new algorithm for calculating Adomian polynomials[J]. Appl Math Comput, 2005, 169:402-416. http://www.sciencedirect.com/science/article/pii/S0096300304007829
    [7] DUAN J S.Convenient analytic recurrence algorithms for the Adomian polynomials[J].Appl Math Comput, 2011, 217:6337-6348. http://www.sciencedirect.com/science/article/pii/S0096300311000117
    [8] DUAN J S, RACH R. The degenerate from of the Adomian polynomials in the power series method for nonlinear ordinary differential equations[J]. J Math System Sci, 2015(5):411-428. http://www.m-hikari.com/nade/nade2017/1-4-2017/p/almuhalbediNADE1-4-2017.pdf
    [9] RACH R, WAZWAZ A M, DUAN J S.A reliable modification of the Adomian decomposition method for higher-order nonlinear differential equations[J].Kybernetes, 2013, 42:282-308. doi:  10.1108/03684921311310611
    [10] LESNIC D.The decomposition method for nonlinear, second-order parabolic partial differential equations[J]. Int J Comput Math Numeric Simul, 2008(l):207-233.
    [11] PATEL A, SERRANO S E.Decomposition solution of multidimensional groundwater equations[J]. J Hydrology, 2011, 397:202-209. doi:  10.1016/j.jhydrol.2010.11.032
    [12] RAMANA P V, RAGHU-PRASAD B K. Modified Adomian decomposition method for Van der Pol equations[J]. International Journal of Non-Linear Mechanics, 2014, 65:121-132. doi:  10.1016/j.ijnonlinmec.2014.03.006
    [13] CHEN F, LIU Q Q. Modified asymptotic Adomian decomposition method for solving Boussinesq equation of groundwater flow[J]. Applied Mathematics and Mechanics, 2014, 35(4):481-488. doi:  10.1007/s10483-014-1806-7
    [14] BOUGOFFA L, RACH R, WAZWAZ A M, et al. On the Adomian decomposition method for solving the Stefan problem[J]. International Journal of Numerical Methods for Heat & Fluid Flow, 2015, 25(4):912-928. doi:  10.1108/HFF-05-2014-0159
    [15] KHAN Y, VAZQUEZ-LEAL H, FARAZ N. An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations[J]. Applied Mathematical Modelling, 2013, 27:2702-2708 http://www.sciencedirect.com/science/article/pii/S0307904X12003824
    [16] HOSSEIEN M M, JAFARI M. A note on the use of Adomian decomposition method for high-order and system of nonlinear differential equations[J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14:1952-1957. doi:  10.1016/j.cnsns.2008.04.014
    [17] SHEHAHA M M. A study of some nonlinear partial differential equations by using Adomian decomposition method and variational iteration method[J]. American Journal of Computational Mathematics. 2015(5):195-203. http://www.oalib.com/paper/3147890
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出版历程
  • 收稿日期:  2017-07-26
  • 刊出日期:  2018-07-25

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