非线性常微分方程数值解的渐近表示以及应用

On asymptotic representation of numerical solutions of the nonlinear order differential equations and its application

对于非线性常微分方程一般不存在解析解,但是通过数值方法发现,有些非线性常微分方程的振荡渐近解是有规律的.因此,可以用最小二乘法等方法对这些数值解拟合出渐近解,在此基础上,再通过理论分析得出更具体的结果,为非线性微分方程的研究提供了一种途径.为了提高计算精度、避免计算过程出现崩溃,我们引入了数值解的函数变换和自变量变换的方法,这也保证了数值结果的可靠性.本文通过对数值解的渐近表示,验证了Painlevé方程振荡渐近解的一些现有结果,并得出一些新的结果.

For the general nonlinear order differential equations one can＇t get analytic solution, but one can find that some oscillating asymptotic solutions is regular by the numerical method. Therefore, for the numerical solutions we can fit to the asymptotic solutions using least square method etc. On this foundation, then we can get a more concrete result by theoretical analysis, this provides a way for the research of the nonlinear order differential equations. In order to improve accuracy and avoid the breakup while computing, we introduce the function transformation and variable transformation of the numerical solution into it, and this also guarantees dependability of the numerical result. In this paper, we verify some existed results of oscillating asymptotic of Painlev equation, and also get a few new results by asymptotic of the numerical solution.