常微分方程的初值问题

常微分方程的初值问题
常微分方程的初值问题的有关性质及其数值解法

摘 要：本文主要讨论了常微分方程的初值问题，针对当前初值问题在常微分方程领域的重要性不断扩大，对初值问题做了1些总结，归纳和介绍。文章主要分3大块，分别是：1：常微分方程的初值问题的有关概念。对涉及到初值问题的1些基本概念做了1些总结。目的就是为了让读者首先对初值问题有个基本的了解。2：常微分方程的初值问题的有关性质。在这里主要介绍了初值问题的5个方面的性质，分别在这几个方面中总结了以下内容：初值问题是否有解，解是否惟1，解的存在区间有多大；初值问题在什么条件下有解----柯西-皮亚诺存在定理；怎样把在小区间上有定义的解开拓到较大的区间上去；初值问题解的惟1性条件---李普希茨条件；3：常微分方程初值问题的数值解法。在这1块主要介绍了欧拉法及龙格－库塔方法。本文对常微分方程的初值问题做了1个比较系统的分析介绍，不足之处，有待以后进1步的研究。
关键词：存在性；开拓性；唯1性；欧拉公式；龙格－库塔公式.
 

The initial value problems of ordinary differential equations
The initial nature and numerical solution of initial value problems of ordinary differential equations

Abstract: This paper mainly discusses the initial value of ordinary differential equations, In light of the initial value problems of ordinary differential equations area of importance is expanding, do some summing up, summarized and presentations about the Initial problems. The article is divided into three major regions, Respectively：1、the initial concept of initial value problems of Ordinary Differential Equations. The initial value problems involving the basic concepts of a conclusive. Its purpose is to give readers a basic understanding  about  the  initial value problems.2、the nature of initial problem of the Ordinary Differential Equations . Here mainly introduces five  natures of  the initial value problems, In several aspects summed up as the following : whether there are  solutions to the problems, whether  the only solution, the existence of solutions interval extent; Initial issues under what conditions have a solution--Cauchy - Piano existence theorem; how the district between the definition of a solution to open up a larger interval up; initial value problems for the sole condition --- Lipschitz conditions; 3、the numerical solution of the initial value problems of Ordinary Differential equations . With the introduction of a major Euler method and Runge - Kutta method. This paper make a  systematic analysis about ordinary differential equation initial value problems.the deficiencies, After pending further study.

Key words: existence; pioneering; uniqueness; Eulers formula; the Runge - Kutta formula .

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