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博碩士論文 etd-0620108-143959 詳細資訊
Title page for etd-0620108-143959
論文名稱
Title
2D Helmholtz 方程之基本解法
The Method of Fundamental Solutions for 2D Helmholtz Equation
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
46
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2008-05-22
繳交日期
Date of Submission
2008-06-20
關鍵字
Keywords
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
the method of fundamental solutions, Bessel functions, Helmholtz equation, error analysis, the method of particular solutions, stability analysis, Neumann functions
統計
Statistics
本論文已被瀏覽 5724 次,被下載 1600
The thesis/dissertation has been browsed 5724 times, has been downloaded 1600 times.
中文摘要
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。
Abstract
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.
目次 Table of Contents
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
參考文獻 References
[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
[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.
[4] Z.C. Li, Error analysis of the Trefftz method for solving Laplaces eigenvalue problems, Journal of Computational and Applied Mathematics, Vol. 200, pp. 231-254, 2007.
[5] Z.C. Li, The Trefftz method for the Helmholtz equation with degeneracy, Applied Numerical Mathematics, vol. 58, 131-159, 2008.
[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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