【作者】 杨丽伟；

【导师】 林福荣；

【作者基本信息】 汕头大学 ， 应用数学， 2008， 硕士

【摘要】 本文在深入了解国内外对Duffing方程周期解存在的充分条件和周期解的解法的基础上,重点对时滞Duffing型方程周期解存在唯一的充分条件和数值解法进行讨论,并给出数值例子.然后简单介绍了几个类似方程周期解存在的充分条件和数值解法.最后,研究了标准形式的Duffing型方程,并给出了一种有效的数值解法.本文分为三章,主要内容如下:第一章,从动力学角度介绍Duffing型方程的产生过程,并介绍一些重要研究成果.第二章,首先介绍时滞Duffing型方程:ax"(t) + bx(t) + g(x(t-τ)) = f(t)的物理背景.接着详细讨论了该方程周期解存在唯一的充分条件,给出了一种数值解法并用例子验证该数值方法的有效性.然后就例子问题还详细讨论了采用不同的初值问题和不同的迭代法来求其数值解.最后简单讨论了下列几种形式的方程:ax"(t) + cx’(t) + bx(t) + g(x(t -τ)) = f(t),ax"(t) + cx’(t) + bx(t) + g(x(t -τ(t))) = f(t),ax"(t) + f[x’(t)] + bx(t) + g(x(t -τ(t))) = p(t)周期解存在的充分条件和数值求解方法.第三章,讨论标准的Du?ng型方程:u"(t) + grad(G(u(t))) = p(t)的数值解.这里我们推广第二章提出的数值方法,并用二维和三维形式的标准Duffing型方程实例验证数值解法的有效性.更多还原

【Abstract】 In the thesis, based on knowing about the su?cient conditions for the exis-tence and uniqueness of periodic solution and numerical methods of di?erentialequation at home and abroad, we consider the existence and numerical solutionmethods of the delay Du?ng di?erential equation. Then, we brie?y introducethe su?cient conditions and numerical solution methods for several kinds ofsimilar equations. Finally we discuss methods of solving the standard Du?ngequation.This thesis consists of three chapters. The main contents are as follows:In chapter 1, we mainly introduce the development of Du?ng equation andreview some significant research.In chapter 2, we first introduce the following delay Du?ng di?erentialequationax" (t) + bx(t) + g(x(t -τ)) = f(t).Here a and b are constants, g(x) and f(t) are functions defined in R with f(t)T-periodic. We introduce the background of delay Du?ng equation and discussthe su?cient conditions for the existence and uniqueness of periodic solution.Then we derive an e?cient numerical method for solving delay Du?ng equa-tion. Numerical examples are given to illustrate the e?ciency of our algorithms.Finally we brie?y discuss su?cient conditions for the existence and uniquenessof periodic solution of the following equations:ax" (t) + cx’ (t) + bx(t) + g(x(t -τ)) = f(t),ax" (t) + cx’ (t) + bx(t) + g(x(t -τ(t))) = f(t),ax" (t) + f[x’(t)] + bx(t) + g(x(t -τ(t))) = p(t).In chapter 3, we discuss a kind of standard Du?ng equation:u (t) + grad(G(u(t))) = p(t) And discuss the numerical solution method of the equation. We extend thenumerical method discussed in the second chapter. Numerical examples of two-dimensional and three-dimensional are given to illustrate the e?ciency of thealgorithm.更多还原