在目前，大部份的文獻都僅討論使用一般解法(MFS)處理2維問題，為了使一般解法(MFS)有更好的效力，本篇將拓展至處理3維問題。
當3維的一般解基底
Φ(x,y)=1/(4π||x-y||), x,y∈R^3
已知，source點的位置在實際計算中便顯得十分重要。在本篇論文中，我們選定柱形的解域，source點佈於比解域大的柱或球體上。最後將有些數值結果與總結一些有用的結論，而理論分析在將來完成。

For the method of fundamental solutions(MFS), many reports deal
with 2D problems. Since the MFS is more advantageous for 3D
problems, this thesis is devoted to Laplace''s equation in 3D
problems. Since the fundamental solutions(FS)
Φ(x,y)=1/(4π||x-y||), x,y∈R^3
are known, the location of source points is important in real
computation. In this thesis, we choose a cylinder as the solution
domain, and the source points on larger cylinders and spheres.
Numerical results are reported, to draw some useful conclusions.
The theoretical analysis will be explored in the future.

Contents
1 Introduction 4
2 Algorithms of Method of Fundamental Solutions 6
3 Particular Solutions of Laplace’s Equation in Cylindrical Coordinates 9
4 Numerical Results 13
5 Conclusions 19

[1] Atkinson, K. E., The Numerical solution of Integral Equations of the Second Kind, Cambridge
University Press, 1997.
[2] Alves, C. J. S. and Leitao, V. M. A., Crack analysis using an enriched MFS domain decomposition
techniques, Engineering Analysis with Boundary Analysis, Vol. 30, pp. 160–166, 2006.
[3] Alves, C. J. S. and Valtchev, S. S., Comparison between meshfree and boundary element mehtods
applied to BVP’s in domains with corners, Proceedings of the ECCM’06, LNEC, Lisbon, Portugal,
Eds. C. A. Mota Soares et.al., Springer, 2006.
[4] Babuska, I. and Aziz, A. K., Survey lectures on the mathematical foundations of the finite element
method in The Mathematical Foundations of the Finite Element Method with Applica-
tions to Partial Differential Equations (Ed. by A. K. Aziz, pp. 3 - 359, Academic Preess, New
York, 1972.
[5] Betcke, T. and Trefethen, L. N.,Reviving the method of particular solutions , SIAM Review, Vol. 47,
pp. 469-491, 2005
[6] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer.
Anal., Vol. 22, pp. 644–669, 1985.
[7] Christiansen, S., On Kupradze’s functional equations for plane harmonic problems, In R. P. Gilbert
and R. Weinacht(Eds): Function Theoretic Method in Differential Equations, London, Research
Notes in Mathematics, Vol. 8, pp. 205–243, 1976.
[8] Christiansen, S., Condition number of matrices derived from two classes of integral equations, Math.
Meth. Appl. Sci., Vol. 3, pp. 364–392, 1981.
[9] Christiansen, S., On the two method for elimination of non-unigue solutions of an integral equation
of an integral equation with logarithmic kernel, Applicable Analysis, Vol. 13, pp. 1–18, 1982.
[10] Chan, F. C. and Foulser, D. E., Effectively will-condition linear systems, SIAM. J. Stat. Comput.,
Vol 9, pp. 963–969, 1988.
[11] Comodi, M. I. and Mathon, R., A boundary approximation method for fourth order problems, Mathematical
Models and Methods in Applied Sciences, Vol. 1, pp. 437 – 445, 1991.
[12] Christiansen, S. and Hansen, P. C., The effective condition number applied to error analysis of
certain boundary collocation method, J. of Comp. and Applied Math., Vol. 54, pp. 15–36, 1994.
[13] Christiansen, S. and Saranen, J., The conditioning of some numerical methods for first kind boundary
integral equations, Vol. 67, pp. 43–58, 1996.
[14] Chen, J. T., Kuo, S. R., Chen, K. H. and Cheng, Y. C., Comments on “ Vibration analysis of
arbitrary shaped membranes using non-dimensional dynamic influence functions”, J. of Sound and
Vibration, pp. 156–171, 2000.
[15] Chen, J. T. and Chiu, Y. P., On the pseudo-differential operators is the dual boundary inteqral
equations using degenerate kernels and circulants, Engineering Analysis with Boundary Elements,
Vol. 26, pp. 41–53, 2002.
[16] Chen, C. S., Cho, H. A. and Golberg, M. A., Some comments on the ill-conditioning of the method
of fundamental solutions, preprint, Department of Mathematics, University of Southern Mississippi,
Hattiesburg, MS 39406, USA, 2005.
[17] Chen, C. S., Hon, Y. C. and Schaback, R. A., Scientific Computing with Radial Basis Func-
tions, preprint, Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS
39406, USA, 2005.
[18] Davis, P. J., Circulant Matrices, John Wiley & Sons, New York, 1979.
[19] Davis, P. J. and Rabinowitz, P., Methods of Numerical Integration (Sec. Ed.), Academic
Press, pp. 315, 1984.
[20] Fairweather, G. and Karageorghis. A., The method of fundamental solurions for elliptic boundary
value problems , Advances in Computional Mathematics, Vol. 9, pp. 69-95, 1998.
[21] Gradsheyan, I. S. and Ryzhik, Z. M., Tables of Integrals, Series and Products, Academic Press,
New York, 1965.
[22] Huang, H. T, Li, Z. C. and Yan N., New error estimates of Adini’s elements for Poisson’s equation,
Applied Numer. Math., Vol. 50, pp. 49 – 74, 2004.
[23] Huang, H. T, Li, Z. C. and Huang, H. T., Stability analysis for the first kind boundary integral
equations by methanical quadratare methods, Technical report, Department of Applied Mathematics,
National Sun Yet-sen University, Kaohsiung, 2006.
[24] Kupradze, V. D., Pontential methods in elasticity, in J. N. Sneddon and R. Hill (Eds), Progress in
Solid Mechanics, Vol. III, Amsterdam, pp. 1-259, 1963.
[25] Kitagawa, T., Asymototic stability of the fundamental solution methods, J. Comp. and Appl. Math.,
vol. 38, pp. 263 – 269, 1991.
[26] Karageorghis, A., Modified methods of fundamental solutions for harmonic and bi-harmonic problems
with boundary singularities, Numer. Meth. for PDE, Vol. 8, pp. 1–18, 1992.
[27] Katsurada, M. and Okamoto, H., The collocation points of the method of fundamental solutions for
the potential problem, Computers Math. Applics, Vol. 31, pp. 123–137, 1996.
[28] Kuo, S. R., Chen, T. J. and Huang, C. X., Analytical study and numerical experiments for true
and spurious eigensolutions of a circular cavity using the real-part deal BEM, Inter. J. for Numer.
Methods in Engrg. Applics, Vol. 84, pp. 1401–1422, 2000.
[29] Karageorghis, A., The method of fundamental solutions for the calculation of the eigenvalues of the
Helmoholtz equation, Applied Mathematics Letter, Vol. 14, pp. 837–842, 2001.
[30] van Loan, C. F., Generalizing the singular value decomposition, SIAM. J. Numer. Anal., Vol. 13, pp.
76–83, 1976.
[31] Liem, C. B., L‥u, T. and Shih, T. M., the Splitting Extrapolation Method, a New Technique
in Numerical Solution of Mathidimensional problems, World Scientfic, Singapore, 1995.
[32] Li, Xin, Convergence of the method of fundamental solutions for solving the boundary value problem
of modified Helmholtz equation, Applied Mathematics and Computation, Vol.159, pp. 113 – 125,
2004.
[33] Li, Xin, On convergence of the method of findamental solutions for solving the Dirichlet problem of
Poisson’s equation, Advances in Computational Mathematics, Vol. 23, pp. 265 – 277, 2005.
[34] Li, Xin, Convergence of the method of fundamental solutions for Poisson’s equation on the unit
sphere, Advances in Computational Mathematics, Vol. 28, 269 – 282, 2008.
[35] Li, Z. C., Combined Methods for Elliptic Equations with Singularties, Interface and
Infinities, Academic Publishers, Dordrecht, Boston and London, 1998.
[36] Lu, T. T., Hu, H. Y. and Li, Z. C., Highly accurate solutions of Motz’s and cracked-beam problems,
Engineering Analysis with Boundary Elements, Vol. 28, pp. 1378-1403, 2004.
[37] Li, Z. C., Huang, H. T. and Chen, J. T., Effective condition number for collocation Trefftz method,
Technical report, Department of Applied Mathematics, National Sun Yet-sen University, Kaohsiung,
2005.
[38] Li, Z. C., Lu, T. T., Hu, H. T. and Cheng, A. H.-D., Trefftz and Collocation Methods, MIT,
Southampton, 2008.
[39] Li, Z. C. and Huang, H. T., Effective condition number for numerical partial differential equations,
Numerical Linear Algebra with Applications, Vol. 15, pp. 575 – 594, 2008.
[40] Li, Z. C., Chien, C. S. and Huang, H. T., Effective condition number for finite difference method, J.
Comp. and Appl. Math., Vol. 198, pp. 208 - 235, 2007.
[41] Li, Z. C., Combinations of Method of fundamental solutions for Laplace’s equation with singularities,
Enguneering Analysis with Boundary Elements, Vol. 32, pp.856 – 869, 2008.
[42] Mathon, R. and Johnston, R. L., The approximate solution of elliptic boundary-value problems by
fundamental solutions, SIAM J. Numer. Anal., Vol. 14, pp. 638–650, 1977.
[43] Mey, G. De., Integral equations for potential problems with the source function not located on the
boundary, Computer & Structure, Vol. 8, pp. 113–115, 1978.
[44] Pinsky, Mark A., Partial Differential Equations and Boubdary-Value Problems with Applications,3rd
, McGraw-Hill, pp. 171-233, 1998.
[45] Oden, J. T., and Reddy, J. N., An Introduction to the Mathematiced Theory of Finite
Elements, John Wiley & Sons, New Yark, 1976.
[46] Ramachandran, P. A., Method of fundamental solutions: Singular value decomposition analysis,
Commum. Method in Engrg., Vol. 18, pp. 789–801, 2002.
[47] Sidi, A., Practical Etropolution Methods, Theory and Applications, Cambridge University
Press, 2003.
[48] Schaback, R., Adaptive numerical solution of MFS systems , a plenary talk at the first Inter. Workshop
on the Method of Fundamental Solutions(MFS2007), Ayia Napa, Cyprus, June 11-13, 2007.
[49] Trefftz, E., Konvergenz und Ritz’schen Verfahren, Proc. 2nd Int. Cong. Appl. Mech., Znrch, pp.
131–137, 1926.
[50] William, E. B. and DiPrima, R. C. Elementary differential equations and boundary value prob-
lems,5ed , Wiley, pp. 111-116, 1992.