克氏分析之邊界積分方程式

Abstract

對於待求場量為調和函數 (即控制方程式為拉普拉斯方程式) 之平面問題, 若以複數形式來呈現, 則在進行解題及分析上極為有效, 因複變分析擁有強大的運算能力及豐富完整的函數論. 有鑑於此, 本論文擬在保有這些特性的情況下, 將之由二維推廣至高維的邊界積分方程式來處理調和函數問題. 考慮到實際問題的需要, 在此推導出的奇異與超奇異邊界積分方程式, 皆適用於任意形狀的邊界問題, 即便是包含角點的邊界. 我們亦證實了實變數、複變數、四元數與克氏值的邊界積分方程式之間存在著關連性, 同時三套克氏值的邊界積分方程式亦可彼此轉換.

It is well known that plane problems of harmonic functions are analyzed and solved effectively when expressed in the form of complex variables. This effectiveness is generally attributed to the powerful techniques of complex analysis and the richness of complex function theory. In view of this, the present thesis is aimed to extend the techniques to n-dimensional problems of boundary integral equations (BIEs) for harmonic field variables. Regarding usefulness for practical purposes, we derive singular and hypersingular BIEs not only for points on smooth boundaries but also for corner boundary points. The relations of real, complex, quaternion, and Clifford valued BIEs are explored. In Clifford valued BIEs, the three types of functions of are treated.