分离变量方法在微分方程求解中的应用分析本科毕业论文.doc

精品文档，知识共享！ 精品文档，知识共享！ 学校代码： 10128学 号： 040905054 学校代码： 10128 学 号： 040905054 本科毕业论文 本科毕业论文 （ 题 目：分离变量方法在微分方程求解中的应用分析 题 目：分离变量方法在微分方程求解中的应用分析 摘 要 分离变量法是求解微分方程的最基本方法之一，对此法进行探索研究具有重要的理论和实践意义. 为了让大家更透彻地理解分离变量法，本文首先在引言中详细地介绍了常微分方程分离变量法的起源及其进展情况；紧接着,在第一章综述了基于Sturm-Liouville问题的传统分离变量法，内容涉及到分离变量法的力学背景、理论基础、基本思想、计算步骤、算例等几方面. 作为对偏微分方程求解方法的补充，我们还就非线性领域的分离变量法做了一个摘要; 最后, 在本文的主体部分—第二章中力图进一步完善辛-Fourier展开法（基于Hamilton体系的分离变量法）的应用体系, 主要工作包括以下三方面：第一，在Hamilton体系下尝试求解常微分方程并验证了该解法的正确性；第二，利用辛-Fourier展开法在Hamilton体系下求出了二阶椭圆型方程的通解,这比前人所做的工作都完善；第三，利用辛-Fourier展开方法初步探讨了抛物型方程，虽然没有得出满意的结果，但是也取得了一些收获. 关键词：传统分离变量法；Sturm-Liouville问题；Hamilton系统；辛空间； 辛-Fourier展开法 Abstract??? The method of Separation of variables is one of the most basic methods to solve differential equations. It provides importantly theoretical and practical significance to exploit and research this method. To make us understand about method of separation of variables, firstly, in the introduction, we study the origin and progress of this method for the ordinary differential equations; after that, we investigate the traditional method of separation of variables based on Sturm-Liouville problem in 1st chapter, which involves the mechanical background, the theoretical foundation, the basic idea, calculated steps, examples and other areas. As a supplementary of the method to solve partial differential equations, we also do a abstract about separation of variables on the non-linear field; finally, in the main part of this paper: 2nd chapter aimed to further improve symplectic-Fourier expansion method (namely, method of separation of variables based on Hamiltonian system) on application, we do the works as following: First, verify the correctness to solve ordinary differential equations in the Hamilton system; Second, take use of symplectic-Fourier expansion method to solve the second-order elliptic equation in Hamiltonian system, the results are more perfect than the work done by their prede