两点边值问题解的一种有效算法

AN EFFICIENT ALGORITHM TWO-POINT BOUNDARY VALUE PROBLEMS

本文讨论非线性微分方程边值问题的数值解。对非线性项在局部予以线性化后,再应用打靶方法求解,可加快收敛过程;同时对线性化的函数值采取插值逼近,进一步减少了计算量。本文算法格式简便、编程容易。若辅助内、外存交换技术,利用本文算法,可在微机上完成较大规模复杂问题的分析。算例表明,本文算法大大快于用牛顿法求解一些差分格式方程的收敛速度。

This paper deals with the numerical solutions for nonlinear boundary value problems governed by ordinary differential equations. The nonlinear functions are locally linearized sequentially and a shooting method is then applied to the linearilized equations.This technique is found to be faster in numerical convergence compared with a standard shooting method (i. e. when it is applied directly to the original nonlinear equations). A further reduction in computer time expenditure can be made with the utilization of interpolation to the Jacobian matrices. The present method is simple and easy to be programmed. By making use an exchange process technique of the internal storage requirements with the external equipments of a machine, one may analysis, with the present method, a rather large and complicated problem in a personal computer . A numerical example shows that the present method requires much fewer iterative steps to reach convergence than some finite difference scheme combined with Newton's method does.