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研究生:羅琳峰
研究生(外文):Lin-Feng Lo
論文名稱:2DHelmholtz方程之基本解法
論文名稱(外文):The Method of Fundamental Solutions for 2D Helmholtz Equation
指導教授:呂宗澤
指導教授(外文):Tzon-Tzer Lu
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2008
畢業學年度:96
語文別:英文
論文頁數:46
中文關鍵詞:基本解法(MFS)Neumann函數Bessel函數穩定性分析誤差分析Helmholtz方程特解方法
外文關鍵詞:the method of fundamental solutionsBessel functionsHelmholtz equationerror analysisthe method of particular solutionsstability analysisNeumann functions
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此論文運用Bessel和Neumann函數為基底之基本解法(MFS)去分析2D Helmholtz方程之誤差和穩定性。我們導出在有界的單連通區域之誤差範圍,然而條件數範圍的推導僅限制在圓盤區域上。由實驗中可看出,運用Bessel函數為基底之基本解法比起Neumann函數為基底更有效率。在Bessel函數為基底之基本解法中,有趣的是,源點(source points)的半徑未必大於方程解區域裡的最大半徑r_max 。這違反了在基本解(MFS)裡的一般條件: r_max < R 。最後,利用實驗數據去驗證理論分析和結論。
In the thesis, the error and stability analysis is made for the 2D Helmholtz equation by the method of fundamental solutions (MFS) using both Bessel and Neumann functions. The bounds of errors in bounded simply-connected domains are derived, while the bounds of condition number are derived only for disk domains. The MFS using Bessel functions is more efficient than the MFS using Neumann functions. Interestingly, for the MFS using Bessel functions, the radius R of the source points is not necessarily larger than the maximal radius r_max of the solution domain. This is against the traditional condition: r_max < R for MFS. Numerical experiments are carried out to support the analysis and conclusions made.
1 Introduction 4
2 Algorithms 4
3 Preliminary Lemmas 11
4 Error Analysis of MFS using Bessel functions for Small k 14
5 Stability Analysis for Disk Domains 21
6 MFS Using Neumann Functions 25
6.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.3 Bounds of Condition Numbers . . . . . . . . . . . 29
7 Numerical Experiments 30
8 Concluding Remarks 32
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[2] J.T. Chen, C.S. Wu, Y.T. Lee, and K.H. Chen, On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations, Computers and Mathematics with Applications, Vol. 53, pp. 851-879, 2007.
[3] P.J. Davis, Circulant Matrices, John Weily and Sins, New York, 1979.
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[6] Z.C. Li, T.T. Lu, H.S. Tsai, and A.H.D. Cheng, The Trefftz method for solving eigenvalue problems, Engineering Analysis with Boundary Elements, Vol. 39, pp. 292-308, 2006.
[7] Z.C. Li, T.T. Lu, H.Y. Hu and A.H.D. Cheng, Trefftz and Collocation Methods, WIT Press, Southampton, Jaunnary 2008.
[8] Z.C. Li , H.T. Huang, A.H.D. Cheng and C.S. Chen, Method of Fundamental Solutions and Effective Condition Number, Monograph (in presentation).
[9] Z.C. Li , J. Huang, T.T. Lu, and H.T. Huang, Stability analysis of method of fundamental solution for mixed problems of Laplace’s equation, Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan, 2008.
[10] G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Inc., Englewood Cliffs, 1993.
[11] T. Ushijima and F. Chiba, A fundamental solution method for the reduced wave problem in a domain exterior to a disc, Journal of Computational and Applied Mathematics, Vol. 152, pp. 545-557, 2003.
[12] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge, University Press, 1980.
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