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

华东师范大学学报(自然科学版) ›› 2018, Vol. 2018 ›› Issue (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

楼智美¹, 王元斌², 王鹏³

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

收稿日期:2017-07-26 出版日期:2018-07-25 发布日期:2018-07-19
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
基金资助:
国家自然科学基金（11472177，11772141）

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

LOU Zhi-mei¹, WANG Yuan-bin², WANG Peng³

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:2017-07-26 Online:2018-07-25 Published:2018-07-19

摘要/Abstract

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

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

中图分类号:

O316

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.

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 Sc, 2018, 2018(4): 129-137.

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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

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

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