On the Complete Base for the Steklov Eigenproblems

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基本資料

Project title

Code/計畫編號

MOST108-2221-E019-001

Translated Name/計畫中文名

論Steklov特徵問題的完備基底

Project Coordinator/計畫主持人

Jeng-Tzong Chen

Funding Organization/主管機關

National Science and Technology Council

Co-Investigator(s)/共同執行人

李家瑋

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=13122724

Year

2019

Start date/計畫起

01-08-2019

Expected Completion/計畫迄

31-07-2020

Bugetid/研究經費

1150千元

ResearchField/研究領域

土木水利工程

邊界特徵解的理論是藉由邊界值問題所發展出來的，這理論通常適用於邊界值問題，特徵值隱含於DtN（Dirichlet to Neumann）映射中，DtN 算子也被稱為 Steklov 算子。在本報告中，我們將採用對偶邊界元素法或對偶邊界積分方程針對Steklov 特徵問題來研究，我們將著重於二維拉普拉斯方程問題，首先我們考慮含單圓、橢圓的案例。為了能夠解析推導上述Steklov 的特徵解，二維拉普拉斯方程的閉合型基本解ln(r)將被使用極座標、橢圓座標展開成退化核的形式。我們也探討權重函數如何影響Steklov 的特徵解。本特徵值問題經邊界元素離散化可轉為標準型特徵值問題，而非一般的非線性特徵值問題。最後探討同心圓環的案例，其中同心圓環考慮為三種形式，一種為內外圓皆是DtN條件，另外兩種皆為混合型 Steklov 特徵問題。The theory of the boundary eigensolutions is developed for boundary value problems. It is general for the boundary value problem. The eigenvalue is imbedded in the Dirichlet to Neumann (DtN) map. The DtN operator is called as Steklov operator. In this report, we study the Steklov eigenproblems by using the dual boundary element method/boundary integral equation method (BEM/BIEM). First, we consider a circle and an ellipse. To analytically derive the eigensolution of above shapes, the closed-form fundamental solution of the 2D Laplace equation, ln(r), is expanded into degenerate kernel by using the polar and elliptical coordinates. After the boundary element discretization of the BIE for the Steklov eigenproblem, it can be transformed to a standard linear eigenequation. Instead of the direct-searching scheme for the eigenvalue equation. This problem can be effectively solved by using the dual BEM. Finally, we consider annulus. Not only the Steklov but also the mixed Steklov eigenproblem for annular domains are considered.

Keyword(s)

邊界特徵解
Steklov 特徵問題
對偶邊界元素法
對偶邊界積分方程法
退化核
boundary eigensolution
Steklov eigenproblems
dual boundary element method
dual boundary integral equation method
degenerate kernel

DSpace CRIS

On the Complete Base for the Steklov Eigenproblems

基本資料

Description