【摘要】：本文主要研究了幾類(lèi)具有奇性的奇攝動(dòng)初邊值問(wèn)題的漸近分析,大致分為兩部分內(nèi)容.第一部分研究了數(shù)值積分法求解奇異奇攝動(dòng)兩點(diǎn)邊值問(wèn)題;第二部分研究了具有高階退化根的奇攝動(dòng)微分方程解的漸近分析.第一章介紹了本文的研究背景及國(guó)內(nèi)外研究現(xiàn)狀,簡(jiǎn)要概述了奇異攝動(dòng)問(wèn)題以及奇異攝動(dòng)理論方法的發(fā)展進(jìn)程,并對(duì)本文的研究目的作簡(jiǎn)要介紹.第二章對(duì)本文所要用到的基本知識(shí)作簡(jiǎn)要概述.第三章介紹數(shù)值積分方法求解具有奇性的奇攝動(dòng)二階兩點(diǎn)邊值問(wèn)題,并給出相關(guān)實(shí)例,驗(yàn)證該方法的有效性.第四至六章研究了具有高階退化根的奇攝動(dòng)微分方程解的漸近分析.采用修正的邊界層函數(shù)法獲得具有多區(qū)現(xiàn)象的邊界層刻畫(huà),繼而得出該問(wèn)題的形式漸近解.最后,利用形式漸近解構(gòu)造問(wèn)題的上、下解,并證明該形式漸近解的一致有效性.
[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.
【學(xué)位授予單位】：安徽工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.8

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