Particular Solutions for Polyharmonic and Polyhelmholtz Partial Differential Equations

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Project title

Translated Name/計畫中文名

多項諧和及多項赫姆霍茲偏微分方程式之特解

Project Coordinator/計畫主持人

Chia-Cheng Tsai

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Information Technology,Toko University

Website

https://www.grb.gov.tw/search/planDetail?id=1694845

Year

2008

Start date/計畫起

01-08-2008

Expected Completion/計畫迄

31-07-2009

Bugetid/研究經費

551千元

ResearchField/研究領域

資訊科學--軟體

本計畫擬推導多項諧和及多項赫姆霍茲偏微分方程式之特解。我們所推導之特解，考慮多項式(polynomials)或多項諧和距離函數(polyharmonic splines)在方程式右側，而我們所討論的方程式包含多項諧和及多項赫姆霍茲偏微分方程式。我們都知道，使用胡曼德(Hörmander) 線性方程式理論，我們可以把方程式系統轉換為多項諧和與多項赫姆霍茲偏微分運算子之積。因此，如果我們能有此方程式的特解，我們就可以運用各種邊界數值方法來解方程式的齊次解。這些邊界數值方法包含：邊界元素法、基本解方法、與Trefftz法等等。我們都知道，有許多工程上的問題的控制方程式是多項諧和及多項赫姆霍茲偏微分方程式，比如各種板的力學模式，有時候，也會遇到多個二次之橢圓偏微分方程式，比如多層之含水層，或多項不同孔隙比率之力學模型。這些多個二次之橢圓偏微分方程式，透過胡曼德(Hörmander) 線性方程式理論，也會轉換為多項諧和及多項赫姆霍茲偏微分方程式，因此，我們需要這些特解來補足邊界數值方法的不足，本研究主要就是來補救目前國際上在這方面研究上的不足之處。同時，我們也將把所推導的特解，結合上述的邊界數值方法，進行各種不同的數值實驗，同時比較我們的結果與解析解或文獻上的解答。另外，我們也擬把我們所推導的理論用在前述之工程問題。並把這些結果發表在國際期刊上。 This project plans to derive the particular solutions for the polyharmonic and the products of Helmholtz partial differential operators with polyharmonic splines and monomials right hand side. By the application of the Hörmander linear partial differential operator theory, a large amount of the systems can be reduced to single equation involving the polyharmonic or the product of Helmholtz differential operators. If the inhomogeneous right hand side of these operators can be removed by the method of particular solutions, then boundary-type numerical methods, such as the boundary element method, the method of fundamental solutions, and the Trefftz method, can be applied to solve these differential equations. It is well-known that the polyharmonic and the poly-Helmholtz equations are encountered in certain engineering problems, such as high order plate theory, and systems involving the coupling of a set of second order elliptic equations, such as a multilayered aquifer system, or a multiple porosity system. These coupled systems can be reduced to a single partial differential equation by using the Hörmander operator decomposition technique. The resultant partial differential equations usually involve the polyharmonic or the products of Helmholtz operators. When boundary-type numerical methods are applied to solve these problems, we require the particular solutions for the polyharmonic and the products of Helmholtz partial differential operators with polyharmonic splines and monomials right hand side. Hence it is important to fill this gap in the application of boundary methods to these engineering problems. Also, we will combine the derived analytical particular solutions with several boundary-type numerical methods, such as the boundary element method, the method of fundamental solutions, and the Trefftz method. Furthermore, we will carry out some computational implementations and compare the results to the analytical solutions or the examples in literature. If possible, we will also apply the derived theory to some practical problems. Moreover, the results of the present project will be published in international journals.

Keyword(s)

特解
多項諧和方程式
多項赫姆霍茲方程式
邊界元素法
對偶邊界元素
基本解方法
距離函數
Particular solution
polyharmonic equation
product of Helmholtz equation
boundary element method
dual reciprocity boundary element method
method of fundamental solutions
Trefftz method
radial basis function

DSpace CRIS

Particular Solutions for Polyharmonic and Polyhelmholtz Partial Differential Equations

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