某些常微分方程数值解法与程序实现

西华大学本科毕业论文毕业设计说明书题目：某些常微分方程数值解法与程序实现学院：年级、专业：学生：学号：指导教师：完成日期：西华大学本科毕业论文目录摘要........................................................................................................................4引言........................................................................................................................61准备知识....................................................................................................................61.1微分方程的解..................................................................................................61.2常微分方程的初值问题及其解的存在唯一性..............................................61.3常微分方程的收敛性与稳定性......................................................................71.4MATLAB开发工具简介.................................................................................81.5用户图形界面GUI制作................................................................................82常微分方程数值解....................................................................................................92.1Euler法.............................................................................................................92.2改进Euler法.................................................................................................102.3Runge-Kutta法...............................................................................................113算法及程序流程图..................................................................................................153.1Euler方法.......................................................................................................153.1.1Euler法的算法设计.............................................................................153.1.2Euler法的程序流程图.........................................................................163.2改进Euler法.................................................................................................193.2.1改进Euler法的算法设计...................................................................193.2.2改进Euler法的程序流程图...............................................................203.3Runge-Kutta方法...........................................................................................213.3.1Runge-Kutta法的算法设计.................................................................213.3.2Runge-Kutta法的程序流程图.............................................................234基于MATLAB的程序界面设计...........................................................................264.1用户图形界面GUI说明及详细操作过程..................................................264.2.1Euler向前法.........................................................................................274.2.2Euler向后法.........................................................................................28西华大学本科毕业论文4.2.3改进Euler法.......................................................................................294.2.4二阶Runge-Kutta方法.......................................................................314.2.5三阶Runge-Kutta方法.......................................................................324.2.5四阶Runge-Kutta方法.......................................................................334.2.6综合比较..............................................................................................354.2用户图形界面源程序.....................................................................................365结论...........................................................................................................................42总结与体会..................................................................................................................43谢辞......................................................................................................................43参考文献......................................................................................................................44西华大学本科毕业论文摘要随着社会的不断发展与进步，微分方程逐渐在越来越多的现实生活中得到了非常广泛的应用，但是，我们根据问题建立的微分方程模型中，只有非常少的模型能够得到对应的解析解，因此，通过对常微分方程数值解的剖析，我们能够更加清楚的认识和了解常微分方程。数值分析是通过研究、开发以及分析各种数学问题来得到数值解的算法。本文主要是阐述几种常见的常微分方程数值解的方法，其中包括Euler向前法、Euler向后法、改进欧拉法、Runge-kutta法。介绍了这几种方法的基本原理及推导过程，并且用matlab实现了这些方法，包括它们算法的思想，程序流程以及具体的实现。最后，利用MATLAB中的图形用户界面GUI的功能来实现，让用户操作起来更加的得心应手。关键词：Euler改进欧拉Runge-kuttaMATLAB图形用户界面西华大学本科毕业论文AbstractWiththecontinuousdevelopmentandprogressofsociety,moreandmoregraduallyinthedifferentialequationinreallifehasbeenappliedwidely,butaccordingtothedifferentialequationmodelestablishedinthemodelonlyveryfewcangetthecorrespondinganalyticalsolution,therefore,forwecanknowmoreaboutthedifferentialequation,wemuststudythedifferentialthenumberofequationssolution.Thenumericalanalysisisthroughresearch,developmentandanalysisofvariousmathematicalproblemstoobtainthenumericalsolutionofthealgorithm.Thispaperismainlyfornumericalsolutionsofordinarydifferentialequationsofseveralcommon,includingtheEulerEulerforwardmethodandbackwardmethod,theimprovedEulermethod,Runge-kuttamethodisintroduced.Thebasicprincipleandthederivationofthesemethods,andthesethemethodisimplementedbyMATLAB,includingtheiralgorithm,programflowandimplementation.Finally,theuseof