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二阶常微分方程边值问题的数值解法

摘 要

求解微分方程数值解的方法是多种多样的，它本身已形成一个独立的研究方向，其要点是对微分方程定解问题进行离散化．本文以研究二阶常微分方程边值问题的数值解法为目标，综合所学相关知识和二阶常微分方程的相关理论，通过对此类方程的数值解法的研究，系统的复习并进一步加深对二阶常微分方成的数值解法的理解，为下一步更加深入的学习和研究奠定基础．

对于二阶常微分方程的边值问题，我们总结了两种常用的数值方法：打靶法和有限差分法．在本文中我们主要探讨关于有限差分法的数值解法．构造差分格式主要有两种途径：基于数值积分的构造方法和基于Taylor展开的构造方法．后一种更为灵活，它在构造差分格式的同时还可以得到关于截断误差的估计．在本文中对差分方法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种方法的优缺点进行了细致的比较．在第一章中，本文将系统地介绍二阶常微分方程和差分法的一些背景材料．在第二章中，本文将通过Taylor展开分别求得二阶常微分方程边值问题数值解的差分格式．在第三章中，在第二章的基础上利用Matlab求解具体算例，并进行误差分析．

关键词：常微分方程,边值问题，差分法，Taylor展开，数值解

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The Numerical Solutions of

Second-Order Ordinary Differential Equations

with the Boundary Value Problems

ABSTRACT

The numerical solutions for solving differential equations are various. It formed an independent research branch. The key point is the discretization of the definite solution problems of differential equations. The goal of this paper is the numerical methods for solving second-order ordinary differential equations with the boundary value problems. This paper introduces the mathematics knowledge with the theory of finite difference. Through solving the problems, reviewing what have been learned systematically and understanding the ideas and methods of the finite difference method in a deeper layer, we can establish a foundation for the future learning.

For the second-order ordinary differential equations with the boundary value problems, we review two kinds of numerical methods commonly used for linear boundary value problems, i.e. shooting method and finite difference method. There are mainly two ways to create these finite difference methods: i.e. Taylor series expansion method and Numerical Integration. The later one is more flexible, because at the same time it can get the estimates of the truncation errors. We give 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. In the first chapter, we will introduce some backgrounds of the ordinary differential equations and the difference method. In the second chapter, we will obtain difference schemes of the numerical solutions of the Second-Order ordinary differential equations with the boundary value problems

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through the Taylor expansion. In the third chapter, we using Matlab to solve the specific examples on the basis of the second chapter, and analyzing the errors.

KEY WORDS: Ordinary Differential Equations, Boundary Value Problems, Finite Difference Method, Taylor Expansion, Numerical Solution

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