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摘 要

本文主要研究二阶常微分方程边值问题的数值解法。对线性边值问题，我们总结了两类常用的数值方法，即打靶法和有限差分方法，对每种方法都列出了详细的计算步骤和Matlab程序代码，通过具体的算例对这两类方法的优缺点进行了细致的比较。

关键字：常微分方程边值问题；打靶法；差分法；

ABSTRACT

This article mainly discusses the numerical methods for solving Second-Order boundary value problems for Ordinary Differential Equations. On the one hand, we review two types of commonly used numerical methods for linear boundary value problems, i.e. shooting method and finite difference method. For each method, we give both the exact calculating steps and Matlab codes. Moreover, we compare the advantages and disadvantages in detail of these two methods through a specific numerical example.

Key words：Boundary-Value Problems for Ordinary Differential Equations；Shooting Method；Finite Difference Method；

目 录

TOC \o "1-3" \h \z \u HYPERLINK \l "_Toc262710959" 第一章 引言 PAGEREF _Toc262710959 \h - 1 -

HYPERLINK \l "_Toc262710960" 第二章 二阶线性常微分方程 PAGEREF _Toc262710960 \h - 3 -

HYPERLINK \l "_Toc262710961" 2.1试射法(“打靶”法) PAGEREF _Toc262710961 \h - 4 -

HYPERLINK \l "_Toc262710962" 2.1.1简单的试射法 PAGEREF _Toc262710962 \h - 4 -

HYPERLINK \l "_Toc262710963" 2.1.2 基于叠加原理的试射法 PAGEREF _Toc262710963 \h - 5 -

HYPERLINK \l "_Toc262710964" 2.2 有限差分法 PAGEREF _Toc262710964 \h - 11 -

HYPERLINK \l "_Toc262710965" 2.2.1 有限差分逼近的相关概念 PAGEREF _Toc262710965 \h - 12 -

HYPERLINK \l "_Toc262710966" 2.2.2 有限差分方程的建立 PAGEREF _Toc262710966 \h - 14 -

HYPERLINK \l "_Toc262710967" 2.2.3 其他边值条件的有限差分方程 PAGEREF _Toc262710967 \h - 15 -

HYPERLINK \l "_Toc262710968" 2.2.4 有限差分方程的解法 PAGEREF _Toc262710968 \h - 17 -

HYPERLINK \l "_Toc262710969" 第三章 二阶非线性微分方程 PAGEREF _Toc262710969 \h - 22 -

HYPERLINK \l "_Toc262710970" 3.1基于牛顿迭代法的打靶法 PAGEREF _Toc262710970 \h - 22 -

HYPERLINK \l "_Toc262710971" 3.1.1 第一类边值条件推导 PAGEREF _Toc262710971 \h - 22 -

HYPERLINK \l "_Toc262710972" 3.1.2 其他边值条件的推导 PAGEREF _Toc262710972 \h - 24 -

HYPERLINK \l "_Toc262710973" 3.1.3 算法及程序代码 PAGEREF _Toc262710973 \h - 25 -

HYPERLINK \l "_Toc262710974" 3.2 基于改进的牛