摘要在自然科学研究和社会生产实践过程中，我们往往通过建立微分方程模型来描 述所遇到的现象与问题，而非线性模型往往比线性模型更能反映运动过程的本质， 非线性模型往往更为复杂，多数情况并不能给出解的解析表达式，甚至无法求得解 析解，因此只能通过数值方法寻找它的近似解。79133

本文对现有数值方法以及非线性方程的一般求解方法进行总结研究，融合 Matlab 程序进行综合比较，对比各个方法的优缺点，系统分析求解非线性问题的方 法。文章开始以非线性模型的初值问题描述为引，阐释数值求解的基本思想，进而 在第二章介绍单步法中的 Euler 法、向后 Euler 法、梯形法、RK 方法以及多步法中 的 Adams 显式方法、Adams 隐式方法，对其中的一些方法做了简单的稳定性分析， 并通过误差精度的比较，揭示方法的可行性；第三章又介绍了非线性方程的一般求 解方法：二分法、迭代法。本文还给出了一些常用方法的 Matlab 程序，方便读者的 学习与理解，对于一些需要对比的例子，还给出了 Matlab 绘制图像，使结论更加清 晰明了。

非线性模型的研究一直都一大热点，探索更多的求解方法将是我们努力的方向。 这样对于实际生活中出现的问题，我们便可以更加方便地通过建立非线性的模型使 问题得到解决。

毕业论文关键词：非线性模型；初值问题；数值解法；迭代法；稳定性

Abstract In the course of natural science research and social production practice, we often describe the phenomenon and problems that we have encountered by establishing the differential equation model。 Nonlinear models tend to react more to the nature of the moving process than the linear model, but nonlinear model is often more complex。 In most cases,the analytical expression of the solution can not be given, and even the analytical solution can not be obtained。 Therefore, it can only find its approximate solution by numerical method。

In this paper, the existing numerical methods and the general solution of nonlinear equations are summarized and studied。 Combined with MATLAB program to carry out a comprehensive comparison, compare the pros and cons of each method and analyze method for solving nonlinear problems by the numbers。 The paper begins with the description of the initial value problem of the nonlinear model, and expound the basic idea of numerical solution。 Further in the second chapter, we introduce  a-single-step method that includes Euler Method, Backward Euler Method, Ladder-Shaped  Method,  RK Method and multistep method, like Adams-Explicit Method and Adams-Implicit Method。 Some of these methods are analyzed, and the feasibility of the method is revealed by comparing the error precision; In the third chapter, the general solution of nonlinear equation is introduced。 This paper also gives some common methods of Matlab program to facilitate the reader's learning and understanding。 For some examples of contrast, I draws the image with the Matlab that makes the conclusion more clear。

The research of nonlinear model has always been a hot topic, and it will be the direction of our efforts to explore more solving methods。 In this way, we can more easily solve the problem by establishing the nonlinear model。

Keywords: Nonlinear Model; Initial Value Problem; Numerical Solution; Iterative Method; Stability

目 录

第一章 绪论 1

1。1 引言 1

1。2  非线性常微分方程初值问题描述 2

1。3  数值求解的基本思想 2

1。4 数值方法的分类