华东师范大学学报（自然科学版） >

2018 , Vol. 2018 >Issue 4: 129 - 137

DOI: https://doi.org/10.3969/j.issn.1000-5641.2018.04.013

物理学与电子学

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
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  • 1. 绍兴文理学院 物理系, 浙江 绍兴 312000;
    2. 绍兴文理学院 数学系, 浙江 绍兴 312000;
    3. 济南大学 土建学院, 济南 250022
楼智美,女,教授,主要从事分析力学研究.E-mail:louzhimei@usx.edu.cn

收稿日期: 2017-07-26

  网络出版日期: 2018-07-19

基金资助

国家自然科学基金（11472177，11772141）

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

  • LOU Zhi-mei ,
  • WANG Yuan-bin ,
  • WANG Peng
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  • 1. Department of Physics, Shaoxing University, Shaoxing Zhejiang 312000, China;
    2. Department of Mathematics, Shaoxing University, Shaoxing Zhejiang 312000, China;
    3. School of Civil Engineering and Architecture, University of Jinan, Jinan 250022, China

Received date: 2017-07-26

  Online published: 2018-07-19

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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.

关键词： Adomian分解法; 二阶非线性微分方程; 近似解析解; 数值解

本文引用格式

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 , 2018(4) : 129 -137 . DOI: 10.3969/j.issn.1000-5641.2018.04.013

Abstract

In a non-inertial rotational reference frame, the motion of a system can be governed by a kind of second-order nonlinear differential equation, in which the numerator and denominator both contain nonlinear terms; in this context, it is hard to obtain an approximate solution for this strongly nonlinear equation. In this paper, we study the approximate solution of the second-order nonlinear differential equation by the Adomian decomposition method. Comparisons between the approximate solution and the numerical solution by using two other methods are also made. The results show that, in the first quarter period, the approximate solutions obtained by the Adomian decomposition method is in good agreement with the numerical solutions and the error of the approximate solutions are smaller than the other solutions obtained by the homotopy asymptotic method.

Key words： Adomian decomposition method; second order nonlinear differential equation; approximate analytical solution; numerical solution

参考文献

[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.
[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.
[4] BIAZAR J, SHAFIOF S M. A simple algorithm for calculating Adomian polynomials[J]. Int J Contemp Math Sci, 2007, 20(2):975-982.
[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.
[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.
[7] DUAN J S.Convenient analytic recurrence algorithms for the Adomian polynomials[J].Appl Math Comput, 2011, 217:6337-6348.
[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.
[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.
[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.
[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.
[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.
[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.
[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
[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.
[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.
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