臺灣博碩士論文加值系統

在本文中考慮一個已知上半圓過定邊界條件，和未知的下半圓的邊界條件所組成的拉普拉斯反算問題作為題目來探討。本論文討論兩個不同的類型之邊界條件，分別是Dirichlet邊界形式，和Neumann邊界形式，將兩種邊界形式運用在未知下半圓上，值得注意的是這兩個不同的邊界形式可分別推導出相同形式的數學式。
然後，本論文運用配點法分別結合這兩個傅立葉展開式，試圖去求解Cauchy反算問題。並分別運用配點法、Galerkin法、正則化積分方程去求解Cauchy反算問題。最後，本文將以一些計算例子說明配點法的計算成果，並指出哪一個計算方法在計算柯西過定半圓邊界條件下的拉普拉斯方程時較為精準。

We consider the inverse problem for Laplace equation by recovering boundary values on an inaccessible part of the circle from an available overdetermined data on an accessible part of the circle. We discuss two different formulations, one is recovering the Dirichlet data and another one is recovering the Neumann data on an inaccessible part. We should notice that these two formulations can be unified into a single integral equation. .
Then we apply the collocation method to solve the inverse Cauchy problem. We separately apply the collocation method, Galerkin method, and a regularized integral equation method to solve Cauchy problem. Finally, numerical examples are used to access the numerical method, which will show the effectiveness of the collocation method in providing excellent estimates of the unknown data from the given data.

摘要
Abstract
目錄
表目錄
圖目錄
第一章 緒論
1.1前言
1.2文獻回顧
1.3研究動機與目的
1.4本文架構
第二章 理論基礎
2.1反問題定義
2.2分析Cauchy反問題
2.3 積分方程的概念三章
第三章 構想
3.1 Cauchy反問題的架構分析
3.2 The Dirichlet formulation
3.3 The Neumann formulation
第四章 配點法求解Cauchy反問題
4.1 配點法
4.2 共軛梯度法
4.2.1 共軛梯度法原理
4.2.2 共軛梯度法操作步驟
4.3 誤差估測
4.3.1 誤差來源以及種類
4.3.2 量測誤差的模擬
4.4 數值驗證
4.4.1 算例一
4.4.2 算例二
4.5 結論
第五章 Galerkin method 求解Cauchy 反問題
5.1 Galerkin method
5.2 結論
第六章 正則積分方法求解Cauchy 反問題
6.1 兩點邊界值
6.2 半解析解
6.3 結論
第七章 結論與未來展望
7.1 結論
7.2 未來展望
參考文獻

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