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研究生:陳世昌
研究生(外文):Shi-Chang Chen
論文名稱:拉普拉斯方程過定邊界值反問題
論文名稱(外文):The inverse problems for overspecified boundary values of Laplace equation
指導教授:劉進賢
指導教授(外文):Chein-shan Liu
學位類別:碩士
校院名稱:國立臺灣海洋大學
系所名稱:機械與機電工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2006
畢業學年度:94
語文別:中文
論文頁數:65
中文關鍵詞:拉普拉斯方程反算問題積分方程正則化羅賓傳導係數裂縫位置
外文關鍵詞:Laplace equationInverse problemFredholm integral equationLavrentiev regularizationRobin coefficientcrack position
相關次數:
  • 被引用被引用:2
  • 點閱點閱:221
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  • 下載下載:15
  • 收藏至我的研究室書目清單書目收藏:0
摘要
本文為發展ㄧ正則化積分方程數值新方法求解拉普拉斯方程式Cauchy反算問題及Robin反算問題。
考慮拉普拉斯方程反算問題,藉由量測邊界上之有用過定邊界條件以恢復估測物難以量測邊界上的邊界值。透過易量測邊界上所得量測條件來計算反問題,則數值結果可應用於求Robin類型反算問題、Cauchy類型反算問題、檢驗圓盤內部裂縫發生位置、零電位未知曲線形狀問題等。
首先,透過傅立葉級數建構出未知函數 的第一類Fredholm積分方程,接著考慮ㄧLavrentiev正則化,是為增加一項 以得出第二類Fredholm積分方程,且由於核函數的逐項分離性質,使得我們可推導求得未知邊界條件 的閉合解。因此,我們可以證明正則化積分方程數值解 為均勻收斂且滿足誤差估測條件。應用此數值新方法於Cauchy反算問題、Robin反算問題、零電位未知曲線形狀問題、圓盤內部裂縫發生問題。透過上述數值算例驗證,顯示此數值新方法效能由已給定邊界條件來求解未知邊界條件時具有相當良好的估測。
We consider a new method that it is developed to solve inverse Cauchy problems and inverse Robin problems for the Laplace equation, which is named the regularized integral equation method (RIEM). The inverse problem for the Laplace equation by recoverning boundary values on the inaccessible boundary of the body from available overspecified data on the accessible boundary. The numerical results can be used to determine the Robin type inverse problem, the Cauchy type inverse problem, the problem of detecting crack position, the unknown shape of Zero potential problem through the measurements at the accessible boundary.
The Fourier series expansion is used to formulate the first kind Fredholm integral equation for the unknown data on the inaccessible boundary. Then we consider a Lavrentiev regularization, by adding an extra term to obtain a second kind Fredholm integral equation. The termwise separable property of kernel function allows us to obtain a closed-form solution of the missing boundary condition. The uniform convergence and error estimate of the regularization solution are proved. Then We apply this method to the Cauchy type inverse problems, the Robin type inverse problems, the unknown shape of Zero-potential problem, as well as the problem of detecting crack position. These numerical examples show the effectiveness of the new method in providing excellent estimates of the unknown data from the given data.
目錄
誌謝 i
摘要 ii
Abstract iii
目錄 v
圖目錄 vii
第一章 緒論 1
1.1 前言 1
1.2 文獻回顧 1
1.3 研究動機與目的 2
第二章 理論基礎 4
2.1 反問題定義 4
2.2 分析Cauchy與Robin問題 5
2.3 積分方程概念 6
2.4 數值方法 7
第三章 新方法求解拉普拉斯方程Cauchy反問題 9
3.1 環形場域Cauchy問題 9
3.1.1 Cauchy反問題架構分析 9
3.1.2 Fredholm積分方程 9
3.1.3 兩點邊界值問題 12
3.1.4 閉合解推導 13
3.2 誤差估測 15
3.3 數值驗證 21
3.3.1 柯西型式反算問題 21
3.3.2 零電位曲線 22
3.3.3 決定裂縫發生位置 24
3.3.4 Robin交換係數 25
第四章 新方法求解拉普拉斯Robin方程反問題 28
4.1 條狀場域Robin問題 28
4.1.1 Robin反問題架構分析 28
4.1.2 Fredholm積分方程 29
4.1.3 兩點邊界值問題 30
4.1.4 閉和解推導 31
4.2 誤差估測 33
4.3 數值驗證 35
4.3.1 考慮 時 37
4.3.2 考慮 為一不平滑函數時 39
第五章 結論與未來展望 42
參考文獻 43
附圖 46
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