基於傳統柯西(Cauchy)積分公式的傳統複變數邊界積分方程求解二維問題是非常合適且強而有力，尤其當未知函數是一個複數值的全純函數。換句話說，此未知函數需滿足柯西-黎曼(Cauchy-Riemann)方程式。然而大部分的實際工程問題是三維問題且不一定滿足柯西-黎曼方程式，因此本論文有兩個目標，其中一個是延伸傳統複變數邊界積分方程求解未知函數為非複數值全純函數的二維問題。另一個則是推導出即使在三維空間仍保有一些複變數特性的廣義邊界積分方程。對於延伸傳統複變數邊界積分方程，本文使用Borel-Pompeiu公式來導得廣義複變數邊界積分方程，以這種方式扭轉問題能以兩個剪應力場為狀態函數來直接求解，此外扭轉剛度也可一併求得。由於複變函數受限於僅適用於二維的問題，本文引入克氏(Clifford)代數與克氏(Clifford)分析來取代複變數處理三維問題，克氏代數可以看成複變數或者是四元數的延伸，且克氏分析也可稱之為超複變分析。本文利用了克氏代數值的斯托克斯(Stokes)定理推導出含柯西(Cauchy)型核函數的克氏代數值邊界積分方程，如此一來，一些含多個未知場量的三維問題可以直接被求解。最後本文則考慮一些電磁波散射問題來檢驗克氏代數值邊界積分方程的正確性。

The conventional complex variable boundary integral equation (CVBIE) based on the conventional Cauchy integral formula is powerful and suitable to solve two-dimensional problems. In particular, the unknown function is a complex-valued holomorphic function. In other words, the unknown function satisfies the Cauchy-Riemann equations. However, the most part of practical engineering problems are three-dimensional problems and do not necessarily satisfies Cauchy-Riemann equations. Therefore, there are two targets in this dissertation. One is to extend the conventional CVBIE to solve two-dimensional problems for which the unknown function is not a complex-valued holomorphic function. The other is to extend to three-dimensions and derive an extended BIE still preserving some properties of complex variables in the three-dimensional state. For the extension of the conventional CVBIE, we employ the Borel-Pompeiu formula to derive the generalized CVBIE. In this way, the torsion problems can be solved in the state of two shear stress fields directly. In addition, the torsional rigidity can also be determined simultaneously. Since the theory of complex variables has a limitation that is only suitable for 2-dimensional problems, we introduce Clifford algebra and Clifford analysis to replace complex variables to deal with 3-dimensional problems. Clifford algebra can be seen as an extension of complex or quaternionic algebras. Clifford analysis is also known as hypercomplex analysis. We apply the Clifford algebra valued Stokes' theorem to derive Clifford algebra valued BIEs with Cauchy-type kernels. In this way, some three-dimensional problem with multiple unknown fields may be solved straightforward. Finally, several electromagnetic scattering problems are considered to check the validity of the derived Clifford algebra valued BIEs.

Contents I
Table captions IV
Figure captions V
Abstract VII
摘要 VIII

Chapter1 Introduction ………………………………………………………….. 1
1.1 Motivation and Literature reviews ………………...………………….. 1
1.2 Organization of the present dissertation …..………………………...... 4
Chapter 2 Generalized complex variable boundary integral equation for stress fields and torsional rigidity of torsion problems…………...... 6
Summary ………………………………………………………………….. 6
2.1 Problem statement ........................................…………………………. 7
2.2 Derivation of the general Cauchy integral formula ..........……………. 9
2.3 Application of general Cauchy integral formula for a torsion bar containing an inclusion .........................................................…............ 11
2.4 Discretization of the complex variable boundary integral equations and matching of boundary conditions ...................................…............ 12
2.5 Numerical examples and discussions .................................................... 18
2.5.1 A circular bar containing an eccentric inclusion ..............………… 18
2.5.2 A circular bar containing an eccentric hole ...................................... 19
2.5.3 A solid torsion bar ………………...............………………………. 19
2.6 Conclusions ……….............…………………………………………... 20
Chapter 3 Boundary integral equations in the Clifford algebra …….....……... 30
Summary ………………………………………………………………….. 30
3.1 Real-valued boundary integral equation in ………………………
30
3.2 Complex variables boundary integral equation in ….......………….
32
3.2.1 Complex differential operators ...............………………………….. 32
3.2.2 Generalized complex variables boundary integral equation in ..
33
3.3 Clifford algebra valued boundary integral equations in ………..
35
3.3.1 Clifford algebra in …...............………………………..
36
3.3.2 Dirac-type operators and Cauchy-type kernels in …….........…
38
3.3.3 Clifford algebra valued boundary integral equation in …..........
39
3.3.3.1 Clifford algebra valued boundary integral equation for the Laplace and the Dirac equations ................................................. 39
3.3.3.2 Clifford algebra valued boundary integral equation for the Helmholtz and the k-Dirac equations .......................................... 41
3.4 Conclusions …………………………………………………………… 42
Chapter 4 Application of the Clifford-valued boundary integral equation to electromagnetic scattering problems .................................................. 46
Summary ………………………………………………………………….. 46
4.1 Problem statement of electromagnetic scattering ..…………………… 46
4.2 Clifford analysis and Clifford algebra valued BIE for electromagnetic scattering problems ………................................……. 48
4.2.1 Clifford analysis for Maxwell's equations …….........…..................... 48
4.2.2 Clifford analysis for the boundary conditions .........................……... 49
4.2.3 Clifford algebra valued boundary integral equation for nonhomogeneous k-Dirac equation ................................................... 50
4.3 Illustrative examples .........……………………………………………. 51
Case 1: An interior problem with a plane wave solution ....……………... 51
Case 2: An exterior problem with a dipole source ….........………....…... 52
Case 3: A concentric spheres with the specified boundary condition …... 53
4.4 Conclusions …………………………………………………………… 54
Chapter 5 Conclusions and future researches …………………………………. 68
5.1 Conclusions …………………………………………………………… 68
5.2 Future researches ……………………………………………………... 69
Appendix A …………………………………………………………………………. 70
Appendix B …………………………………………………………………………. 82
Appendix C …………………………………………………………………………. 83
Appendix D …………………………………………………………………………. 85
References …………………………………………………………………………... 87
作者簡歷 ……………………………………………................................................ 92

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