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博碩士論文 etd-0823110-144420 詳細資訊
Title page for etd-0823110-144420
論文名稱
Title
特解法結合基本解法中形狀參數變化的影響
On the Shape Parameter of the MFS-MPS Scheme
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
37
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
2010-07-23
繳交日期
Date of Submission
2010-08-23
關鍵字
Keywords
特解法、徑向基底函數、譜方法、無網格法、誤差估計、基本解法
particular solution, method of fundamental solution, radial basis functions, error estimate, meshless method, Spectral Collocation method using Polynomial
統計
Statistics
本論文已被瀏覽 5720 次,被下載 1227
The thesis/dissertation has been browsed 5720 times, has been downloaded 1227 times.
中文摘要
在這篇論文中,我們利用一種新的方法來解Poisson equation,它是由結合特解法(MPS)及基本解法(MFS)所成的one-step method,稱為MFS-MPS。在一維的Poisson equation情況下,我們證明出用MFS-MPS法所得到的解會收斂到譜方法(Spectral Collocation Method using Polynomial)的解,而結果類似於用特解法、Kansa’s method、譜方法來解。在二維,我們也測試了幾個微分方程的例子,並觀察其誤差行為。
Abstract
In this paper, we use the newly developed method of particular solution (MPS) and one-stage method of fundamental solution (MFS-MPS) for solving partial differential equation (PDE). In the 1-D Poisson equation, we prove the solution of MFS-MPS is converge to Spectral Collocation Method using Polynomial, and show that the numerical solution similar to those of using the method of particular solution (MPS), Kansa's method, and Spectral Collocation Method using Polynomial (SCMP). In 2-D, we also test these results for the Poisson equation and find the error behaviors.
目次 Table of Contents
1. Introduction (P.4)
2. Radial Basis Function Collocation Method (P.4)
2.1. Radial Basis Functions (P.4)
2.2. Interpolation (P.5)
2.3. Kansa's Method of Partial Differential Equation (P.6)
2.4. MFS-MPS for Partial Differential Equation (P.7)
3. One-Dimensional PDE (P.9)
3.1. Error Measures (P.11)
4. Numerical Result by One-Dimensional PDE (p.12)
4.1. Example 1 (P.13)
4.2. Example 2 (P.16)
4.3. Example 3 (P.19)
5. Two-Dimensional PDE (P.22)
6. Solution of PDE with uniform Grid (P.22)
6.1. Example1 (P.23)
6.2. Example2 (P.25)
6.3. Example3 (P.27)
6.4. Example4 (P.29)
6.5. Example5 (P.31)
7. Conclusions (P.33)
Appendix (P.33)
References (P.34)
參考文獻 References
1. Baxter, B.J.C. The asymptotic cardinal function of the multiquadratic f(r) =(r2 + c2)^(1/2) as c→∞.

2. Bogomolny, A. Fundamental-solutions method for elliptic boundary-value problems.

3. Buhmann, M.D. Radial Basis Functions: Theory and Implementations.

4. Chen, C.S., Fan C.M., Monroe J. The method of fundamental solutions for solving elleptic pdes with variable coefficients.

5. Cheng, A.H.-D. & Cabral, J.J.S.P. Direct solution of ill-posed boundary value problems by radial basis function collocation method.

6. Cheng, A.H.-D., Golberg, M.A., Kansa, E.J., and Zammito, G. Exponential convergence and h-c multiquadric collocation method for partial differential equations.

7. Driscoll, T.A. & Fornberg, B. Interpolation in the limit of increasingly flat radial basis functions.

8. Fornberg, B., Wright, G., and Larsson, E. Some observations regarding interpolants in the limit of flat radial basis functions.

9. Huang, C.-S., Lee, C.-F. & Cheng, A.H.-D. Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method.

10. Kansa, E.J., Multiquadrics|a scattered data approximation scheme with applications to computational fluid-dynamics. 1. Surface approximations and partial derivative estimates.

11. Kansa, E.J. Multiquadrics|a scattered data approximation scheme with applications to computational fluid-dynamics. 2. Solutions to parabolic, hyperbolic and elliptic partial-differential equations.

12. Larsson E. & Fornberg, B. Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions.

13. Micchelli, C.A. Interpolation of scattered data-distance matrices and conditionally positive definite functions.

14. Schaback, R. Multivariate interpolation by polynomials and radial basis functions.

15. Vertnik, R. & Sarler, B. Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems.

16. Wendland, H. Scattered Data Approximation.

17. Yen, H.-D. On the increasingly flat RBFs based solution methods for elliptic PDEs and interpolations.

18. Chen, C.S., Fan C.M., Wen P.H.. The Method of Approximate Particular Solution for Solving Certain Partial Differential Equations.
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