具有奇性的奇摄动问题的渐近分析
发布时间：2019-03-21 12:28
【摘要】：本文主要研究了几类具有奇性的奇摄动初边值问题的渐近分析,大致分为两部分内容.第一部分研究了数值积分法求解奇异奇摄动两点边值问题;第二部分研究了具有高阶退化根的奇摄动微分方程解的渐近分析.第一章介绍了本文的研究背景及国内外研究现状,简要概述了奇异摄动问题以及奇异摄动理论方法的发展进程,并对本文的研究目的作简要介绍.第二章对本文所要用到的基本知识作简要概述.第三章介绍数值积分方法求解具有奇性的奇摄动二阶两点边值问题,并给出相关实例,验证该方法的有效性.第四至六章研究了具有高阶退化根的奇摄动微分方程解的渐近分析.采用修正的边界层函数法获得具有多区现象的边界层刻画,继而得出该问题的形式渐近解.最后,利用形式渐近解构造问题的上、下解,并证明该形式渐近解的一致有效性.
[Abstract]:In this paper, we mainly study the asymptotic analysis of some singularly perturbed initial-boundary value problems, which are divided into two parts. In the first part, the numerical integration method is used to solve the singularly perturbed two-point boundary value problem, and in the second part, the asymptotic analysis of the solution of the singularly perturbed differential equation with higher order degenerate roots is studied. The first chapter introduces the research background of this paper and the research status at home and abroad, briefly summarizes the singular perturbation problem and the development of singular perturbation theory and methods, and gives a brief introduction to the purpose of this paper. The second chapter gives a brief overview of the basic knowledge to be used in this paper. In the third chapter, the numerical integration method is introduced to solve singularly perturbed second-order two-point boundary value problems, and an example is given to verify the effectiveness of this method. In chapters 4 to 6, the asymptotic analysis of solutions of singularly perturbed differential equations with degenerate roots of higher order is studied. The modified boundary layer function method is used to obtain the boundary layer characterization with multi-region phenomenon, and then the formal asymptotic solution of the problem is obtained. Finally, the upper and lower solutions of the problem are constructed by using the formal asymptotic solution, and the uniform validity of the formal asymptotic solution is proved.
【学位授予单位】：安徽工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2444933
[Abstract]:In this paper, we mainly study the asymptotic analysis of some singularly perturbed initial-boundary value problems, which are divided into two parts. In the first part, the numerical integration method is used to solve the singularly perturbed two-point boundary value problem, and in the second part, the asymptotic analysis of the solution of the singularly perturbed differential equation with higher order degenerate roots is studied. The first chapter introduces the research background of this paper and the research status at home and abroad, briefly summarizes the singular perturbation problem and the development of singular perturbation theory and methods, and gives a brief introduction to the purpose of this paper. The second chapter gives a brief overview of the basic knowledge to be used in this paper. In the third chapter, the numerical integration method is introduced to solve singularly perturbed second-order two-point boundary value problems, and an example is given to verify the effectiveness of this method. In chapters 4 to 6, the asymptotic analysis of solutions of singularly perturbed differential equations with degenerate roots of higher order is studied. The modified boundary layer function method is used to obtain the boundary layer characterization with multi-region phenomenon, and then the formal asymptotic solution of the problem is obtained. Finally, the upper and lower solutions of the problem are constructed by using the formal asymptotic solution, and the uniform validity of the formal asymptotic solution is proved.
【学位授予单位】：安徽工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
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