两类奇摄动微分系统解的性质研究

发布时间：2018-04-14 06:06

本文选题：非线性奇异摄动系统 + 初边值问题　；参考：《太原理工大学》2017年硕士论文

【摘要】：本文主要针对两类奇摄动微分系统(时滞微分系统和非时滞微分系统),给出不同系统解的存在性、收敛性等性质.本文主要工作如下:对于一类含有时滞项的奇摄动微分方程的边值问题,核心方法是利用摄动方法中将某一项展开的思想来处理时滞项,并根据边界层位置的不同对方程的解给出了相应的存在性定理;对于另一类不含时滞项的奇异摄动微分方程的初值问题,利用迭代的思想,并结合摄动方法中解的特点,给出一种简单有效的迭代方法来求解方程的近似周期解.然后结合几个典型的例子来说明该方法对解决这类非线性奇异摄动问题的实用性.同时对这类系统应用了窗口技术以加速迭代过程的收敛.并给出应用窗口技术后系统解与原系统解的误差估计表达式.这些方法可以很容易的扩展到其他非线性系统并发现广泛适用于工程问题中.全文结构如下:第一章简要介绍所研究问题的背景及文中用到的一些基础知识,同时给出了本文得到的主要结果.第二章研究一类含时滞项的奇摄动泛函微分方程边值问题的解的存在性.第三章对一类不含时滞项的奇异摄动微分方程的初值问题进行研究.最后在第四章中对文章作出了总结并给出有待研究的内容.
[Abstract]:In this paper, the existence and convergence of solutions for two classes of singularly perturbed differential systems (delay differential systems and non-delay differential systems) are given.The main work of this paper is as follows: for the boundary value problem of a class of singularly perturbed differential equations with time delay, the core method is to deal with the delay term by using the idea of expanding a certain term in the perturbation method.The existence theorem for the solution of the equation is given according to the different location of the boundary layer, and for the initial value problem of another class of singular perturbation differential equations without delay, the idea of iteration is used and the characteristics of the solution in the perturbation method are combined.A simple and efficient iterative method is presented to solve the approximate periodic solution of the equation.Then several typical examples are given to illustrate the practicability of this method for solving this kind of nonlinear singular perturbation problem.At the same time, the window technique is applied to this kind of system to accelerate the convergence of the iterative process.The error estimation expressions of the system solution and the original system solution after the application of window technique are given.These methods can be easily extended to other nonlinear systems and are widely used in engineering problems.The structure of the paper is as follows: in Chapter 1, the background of the problem and some basic knowledge used in this paper are briefly introduced, and the main results obtained in this paper are also given.In chapter 2, we study the existence of solutions for a class of singularly perturbed boundary value problems of functional differential equations with delays.In chapter 3, the initial value problem of a class of singular perturbed differential equations without delay is studied.Finally, in the fourth chapter, the article is summarized and the content to be studied is given.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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