一阶微分方程的MATLAB数值解法

毕业论文题目一阶微分方程的matlab数值解法学院数学科学学院专业数学与应用数学班级数学1303班学生邹健峰学号220130921157指导教师邢顺来讲师二〇一七年五月七日济南大学毕业论文-I-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-Kutta法，介绍以上六种方法的主要内容，简单介绍了MATLAB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法济南大学毕业论文-II-ABSTRACTThenumericalsolutionofdifferentialequationsisahotspotformodernmathematiciansandscientists,theMATLABnumericalsolutionofthedifferentialequationcanhelpussolvemanymathematicalproblemsinreallife,improvetheefficiencyofthecomputertohelpussolvetheproblem.ThispapermainlydiscussesthreekindsofEulermethodsandthreeRunge-KuttamethodsinMATLABnumericalsolutionoffirstorderdifferentialequation,introducethemaincontentsoftheabovesixmethods,abriefintroductiontothedevelopmentofMATLABrelatedcontentanddifferentialequationsofabriefhistory.Theprogrammingoftheabovealgorithmisstudiedbytheconcretedifferentialequation,headvantagesanddisadvantagesoftheabovemethodsareexploredthroughMATBLABsolvingthespecificfirst-orderdifferentialequation,throughthecharttoanalyzetheconclusions:TheimprovedEulermethodandthefourthorderRunge-Kuttamethodhavehigheraccuracy，hasgoodalgorithmstability.Keywords：FirstOrderDifferentialEquation;NumericalSolution;MATLAB;EulerMethod;Runge-KuttaMethod济南大学毕业论文-III-目录摘要........................................................................................................错误!未定义书签。ABSTRACT...........................................................................................错误!未定义书签。1前言....................................................................................................错误!未定义书签。1.1引言..........................................................................................错误!未定义书签。1.2常微分方程的发展简史...........................................................错误!未定义书签。1.3国内外研究现状......................................................................错误!未定义书签。1.4研究主要内容及研究方案......................................................错误!未定义书签。1.4.1研究的主要内容............................................................错误!未定义书签。1.4.2研究方案........................................................................错误!未定义书签。1.5研究难点.................................................................................................................32预备知识...........................................................................................................................42.1显式Euler法..........................................................................................................42.2隐式Euler法..........................................................................................................52.3改进Euler法..........................................................................................................52.4二阶Runge-Kutta法..............................................................................................62.5三阶Runge-Kutta法..............................................................................................72.6四阶Runge-Kutta法..............................................................................................72.7单步法的收敛性和稳定性.....................................................................................93微分方程数值解法的编程实现.....................................................................................113.1常微分方程的符号解法.......................................................................................113.2显式Euler法的编程实现....................................................................................123.3隐式Euler法的编程实现....................................................................................143.4改进Euler法的编程实现....................................................................................173.5二阶Runge-Kutta法的编程实现........................................................................203.6三阶Runge-Kutta法的编程实现........................................................................223.7四阶Runge-Kutta法的编程实现........................................................................24结论.....................................................................................................................................29参考文献.............................................................................................................................30致谢.....................................................................................................................................31附录.....................................................................................................................................32济南大学毕业论文-1-1前言1.1引言JamesBernoulli在1676年写给Newton的信中首次谈到了微分方程。大约到了十八世纪中期，微分方程才成为一门独立的学科，微分方程是研究和揭示自然规律不可或缺的重要工具。众所周知，在1846年数学家和天文学家共同发现的海王星就是微分方程的功劳。还有在1991年，科学考察队在Alps山脉发现了一个冰封保存完好的人类，然后通过仪器测得碳元素衰变的速率，建立微分方程数学模型，得出了这个人是生活在距今5000多年前的时代。Newton、Leibniz、Bernoullifamily、Lagrange、Laplace等著名数学家们都对微分方程的发展做出了巨大的贡献。微分方程的定义：如果知道自变量、未知函数以及函数的导数（或微分）组成的关系式我们就称为微分方程。常微分方