非局部邊界條件之奇異擾動方程的嚴格數學分析

A rigorous analysis for the singularly perturbed problem with nonlocal boundary condition

林鉍偉(Bi-Wei Lin)

指導教授 ： 李俊璋

國立清華大學/理學院/計算與建模科學研究所/碩士(2020年)

摘要

本文探討某類奇異擾動問題，我們考慮一類非局部邊界條件。
我們將研究解函數u在參數趨近於零時所呈現之漸近行為。說的更詳盡一點，我們證明了當參數趨於0時，u將在定義域(0,l)中任意的閉區間內將以指數方式遞減至零。同時，我們也展示了u在x=0與x=l的邊界層行為。更進一步地，我們證明了存在一個正參數使得此方程的解u具有唯一性，並且建立了u在邊界層的梯度估計相對於微小參數的非平凡漸進展開。我們主要的核心策略，就是將原來的非局部邊界條件奇異擾動微分方程解耦成兩個具有局部邊界條件之奇異擾動微分方程，以及應用偏微分方程中的比較定理來處理此問題。

關鍵字

奇異擾動微分方程 ； 非局部邊界條件 ； 邊界層 ； 漸近行為 ； 內層解

並列摘要

This thesis is concerned with a class of singularly perturbed problems in the domain (0,l) with a small parameter and the following nonlocal boundary conditions.
We investigate the asymptotic behavior of solutions approaches to zero. Precisely speaking, we show that as small parameter tends to 0.The function u exponentially decays to zero in any compact subset of (0,l), and exhibits boundary layers near x=0 and x=l. Moreover, there exists small positive parameter such that we obtain the uniqueness of u , and establish the gradient estimate and nontrivial asymptotic expansions of u(0) and u(l) with respect to the small parameter. The main idea is to transform the original equation into the combination of two equations with local boundary conditions and apply the PDE comparison theorem to the singular perturbed problem.

並列關鍵字

Singularly perturbed ODE ； Nonlocal boundary condition ； Boundary layers ； Asymptotic behavior ； Inner solutions

參考文獻

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