本文選題：非線性奇異攝動系統(tǒng) + 初邊值問題��； 參考：《太原理工大學》2017年碩士論文

【摘要】：本文主要針對兩類奇攝動微分系統(tǒng)(時滯微分系統(tǒng)和非時滯微分系統(tǒng)),給出不同系統(tǒng)解的存在性、收斂性等性質.本文主要工作如下:對于一類含有時滯項的奇攝動微分方程的邊值問題,核心方法是利用攝動方法中將某一項展開的思想來處理時滯項,并根據(jù)邊界層位置的不同對方程的解給出了相應的存在性定理;對于另一類不含時滯項的奇異攝動微分方程的初值問題,利用迭代的思想,并結合攝動方法中解的特點,給出一種簡單有效的迭代方法來求解方程的近似周期解.然后結合幾個典型的例子來說明該方法對解決這類非線性奇異攝動問題的實用性.同時對這類系統(tǒng)應用了窗口技術以加速迭代過程的收斂.并給出應用窗口技術后系統(tǒng)解與原系統(tǒng)解的誤差估計表達式.這些方法可以很容易的擴展到其他非線性系統(tǒng)并發(fā)現(xiàn)廣泛適用于工程問題中.全文結構如下:第一章簡要介紹所研究問題的背景及文中用到的一些基礎知識,同時給出了本文得到的主要結果.第二章研究一類含時滯項的奇攝動泛函微分方程邊值問題的解的存在性.第三章對一類不含時滯項的奇異攝動微分方程的初值問題進行研究.最后在第四章中對文章作出了總結并給出有待研究的內容.
[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.
【學位授予單位】：太原理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175

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本文編號：1748025