龙格-库塔法,求解常微分方程

隆格库塔法求解常微分方程

隆格库塔法求解常微分方程

摘 要

科学技术中常常需要求解常微分方程的定解问题,这里问题最简单的形式,是本章将着重考察的一阶方程的初值问题.虽然 求解常微分方程有各种各样的解析方法,但解析方法只能用来求解一些特殊类型的方程,实际问题中归结出来的微分方程主要靠数值解法求解.

本文着重讨论了隆格库塔法求解一阶常微分方程的初值问题,采用了精度较高的经典的四阶隆格库塔法,然后通过对实例运用Matlab编程进行计算求解,为了体现计算结果的精确性和方法的优越性,再采用了欧拉法和预估较正法对实例进行计算求解作为比较.通过比较三种方法的计算精度,发现四阶经典龙格-库塔方法的误差最小,预估较正法其次,欧拉方法误差则比较大.最后通过选取不同的步长,研究了不同的步长对隆格库塔法求解常微分方程初值问题的计算精度的影响.

总之,本文全面分析了隆格库塔法在求解常微分方程的应用,相比与其他的数值解法,隆格库塔法计算精度较高,收敛性较好,其中四阶的隆格库塔法的效率最高,精度也最高.

关键词:四阶隆格库塔法;欧拉法;预估较正法;一阶常微分方程;Matlab

隆格库塔法求解常微分方程

Runge Kutta Method For Solving Ordinary Differential Equations

ABSTRACT

Problem solving ordinary differential equations are often needed in science andtechnology. the problem in the simplest form is the initial value problem of first order equations in this chapter ,which will be discussed. Although there are various analytical methods for solving ordinary differential equations, the analytical method can only be used to solve some special types of equations.differential equations can be summed up the actual problems which This paper discusses the initial value problem of Runge Kutta Barclays by solving a differential equation, using the four order Runge Kutta method with high accuracy.for instance through classic Matlab programming calculation, the superiority in order to accurately and reflect the calculation result, then the Euler method and the prediction correction method for instance by calculation through the calculation precision. The comparison of three kinds of methods, found that the error of four order Runge Kutta method of minimum, prediction correction method secondly, Euler method error is relatively large. Finally, by selecting different step, study the affect the calculation accuracy of different step of Runge Kutta method to solve initial value problems of ordinary differential equations. In short, this paper comprehensively analyzes the application of Runge Kutta method for solving ordinary differential equations, compared with the numerical solution of other, higher accuracy Runge Kutta method, good convergence, the Runge Kutta method of order four of the highest efficiency and its precision is the highest.

Key words: Four order Runge Kutta method; Euler method; prediction correction method;

first order ordinary differential equations; Matlab

隆格库塔法求解常微分方程

目 录

1 问题的提出......................................................1 1.1 问题背景............................................... ....1 1.2 问题的具体内容..............................................1 2 问题假设........................................................2 3 符号系统........................................................2 4 问题的分析......................................................3 4.1 欧拉格式....................................................3 4.2 预估较正法..................................................3 4.3 四阶隆格库塔法的格式........................................4 5 模型的建立与求解................................................4 5.1 隆格库塔法的基本原理........................................4 5.1.1 Taylor级数............................................4 5.1.2 隆格库塔法的基本思想..................................4 5.1.3 四阶的隆格库塔法......................................5 5.2 其他求解常微分方程边值问题算法的简介..........................6 5.3 模型求解.....................................................8 5.3.1 运用MATLAB软件对模型求解结果及析.......................8 6 模型的评价......................................................16 7 课程设计的总结与体会............................................16 参考文献..........................................................17 附录..............................................................18

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