二阶非线性常微分方程的一种改进再生核方法

针对传统再生核方法求解二阶非线性微分方程只是一阶的方法,收敛速度慢的问题.通过研究再生核方法的误差估计,采用外推技巧消去了误差余项中的低阶无穷小量,给出了求解二阶非线性常微分方程初值问题的一种改进的再生核方法,在只增加少量计算的条件下使得收敛速度可以达到至少三阶.减少了原再生核方法的计算量,提高了收敛速度,数值算例表明该方法在求解非线性问题的有效性.

An Improved Reproducing Kernel Method for the Solutions to the Second-Order Nonlinear Ordinary Differential Equations

According to the slow convergence of the traditional reproducing kernel method,which is a first-order method for solving second-order nonlinear differential equation,through the research of the error estimation of the reproducing kernel method,using the extrapolation technique to eliminate the low order infinitesimal in error estimation,an improved reproducing kernel method is given to solve the problems of second-order nonlinear ordinary differential equations with initial values,which convergent speed can reach at least third-order,only adding a small amount of calculation.The calculation of the reproducing kernel method is greatly reduced,the rate of its convergence is improved,and the validity of solving nonlinear problems is verified by numerical examples.