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

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.

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.