一阶微分方程的matlab数值解法.doc

毕业论文

题 目

学 院

专 业

班 级

学 生

学 号

指导教师

二〇一七年月日摘 要

微分方程的数值解法是近现代数学家和科学家们研究的热点，微分方程的MATLAB数值解法可以帮助我们解决现实生活中的许多数学问题，提高计算机帮助我们解决问题的效率。本文主要讨论研究一阶微分方程的MATLAB数值解法中的三种Euler法和三种Runge-KuttaMATLAB的相关内容和微分方程的发展简史。通过具体的微分方程来研究以上算法的编程实现，通过MATBLAB求解具体的一阶微分方程来探究以上方法的优缺点，通过图表来分析得出结论：改进Euler法和四阶Runge-Kutta法的阶的精度较高，具有较好的算法稳定性。

关键词：一阶微分方程；数值解法；MATLAB；Euler法；Runge-Kutta法

ABSTRACT

The numerical solution of differential equations is a hot spot for modern mathematicians and scientists, the MATLAB numerical solution of the differential equation can help us solve many mathematical problems in real life, improve the efficiency of the computer to help us solve the problem. This paper mainly discusses three kinds of Euler methods and three Runge-Kutta methods in MATLAB numerical solution of first order differential equation, introduce the main contents of the above six methods, a brief introduction to the development of MATLAB related content and differential equations of a brief history. The programming of the above algorithm is studied by the concrete differential equation, he advantages and disadvantages of the above methods are explored through MATBLAB solving the specific first-order differential equation, through the chart to analyze the conclusions: The improved Euler method and the fourth order Runge-Kutta method have higher accuracy，has good algorithm stability.

Key words：First Order Differential Equation; Numerical Solution; MATLAB; Euler Method; Runge-Kutta Method

目 录

摘要I

ABSTRACTII

1 前言11.1 引言1

1.22

1.3 国内外研究现状2

1.4 研究主要内容及研究方案3

1.4.1 研究的主要内容3

1.4.2 研究方案3

1.5 研究难点

2 预备知识

2.1 显式Euler法4

2.2 隐式Euler法5

2.3 改进Euler法5

2.4 二阶Runge-Kutta法6

2.5 三阶Runge-Kutta法7

2.6 四阶Runge-Kutta法7

2.7 单步法的收敛性和稳定性9

3 微分方程数值解法的编程实现11

3.1 常微分方程的符号解法11

3.2 显式Euler法的编程实现12

3.3 隐式Euler法的编程实现14

3.4 改进Euler法的编程实现17

3.5 二阶Runge-Kutta法的编程实现20

3.6 三阶Run