臺灣博碩士論文加值系統

在本研究中利用MQ 半徑基底函數法（Multiquadrics Radial Basis Functions Method）解微分方程式，並探討其計算之準確性和適用性。MQ 半徑基底函數中，包含一個形狀參數，此形狀參數之值對整體計算準確性有相當大之影響。當形狀參數增加時，通常可以得到較好之結果，但卻會造成ill-condition 之問題；另一方面，增加佈點點數亦可增加計算之準確性，但有限於目前一般個人電腦，其所能處理矩陣大小之上限約為1500×1500，故在點數增加上有所侷限。
MQ半徑基底函數法，為所謂免切割(mesh-free or meshless )的數值方法，故其可在非規則區間中應用隨機佈點之方式求解。本研究亦探討佈點隨機程度會對整個計算之影響，得到隨機佈點的分散程度越均勻，計算結果越佳；同時在隨機佈點之情況下，其形狀參數如果使用各個參考點跟參考點之間最近之距離，會有較大之誤差產生，而此時如果改用所有最參考點和參考點最近距離之平均值，可以得到較好之結果。
當區間切割應用在此半徑基底函數法時，在形狀參數較小時，可以得到不錯之結果，但因區間與區間之多點重合緣故，使所需計算矩陣之條件數大幅上升，進而導致計算結果發散或無法完全收斂。雖然應用區間切割法，可以突破目前個人電腦所能處理之矩陣大小上限，讓總佈點數再增加，不過依然會有ill-condition 之問題產生。

Several influential factors in solving differential equations by the MQ RBFs method (Multiquardics Radial Basis Functions Method) are investigated in this study.It is found that the accuracy of solutions generally increases with increasing shape parameters contained in the MQ RBFs and the number of node points in the domain.
However, large values of the shape parameter usually lead the method ill-conditioned.In addition, the resultant coefficient matrix becomes unsolvable when the total number of node points exceeds 1500 on a typical personal computer.
MQ RBFs method is a truly meshless algorithm, which has shown to be effective in solving complicated physical problems with irregular domains. A mechanism to allocate the node points in irregular domains is introduced in this study. The ‘randomness’ of the randomly-distributed node points can be adjusted by a b-factor. It
is found that accuracy of solutions increases with decreasing the ‘randomness’ of the distribution of the node points. In addition, the solution accuracy increases further by utilizing improved shape parameters which are estimated by the average distance of the neighboring node points.
Method of overlapping domain decomposition is also incorporated into the MQ RBFs method for solving partial differential equations. Solutions by overlapping domain decomposition can have the same accuracy as in the situation without domain
decomposition for small values of shape parameters. However, for large shape parameters, increasing the overlapping points will cause the matrix of sub-domain ill-conditioned. The merit of overlapping domain decomposition is that problems with large numbers of node points, which are originally unsolvable without domain decomposition, can now be solved by this approach.

第一章..............................................................1
1.1 研究動機........................................................1
1.2 研究目的........................................................1
1.3 組織與章程......................................................2
第二章..............................................................3
2.1 文獻回顧........................................................3
2.1.1 函數近似（Function Approximations）...........................3
2.1.2 解微分方程式..................................................6
2.2 Kansa-MQ 半徑基底函數法.........................................8
第三章............................................................ 11
3.1 規則性佈點之探討.............................................. 11
3.1.1 一維之微分方程式.............................................12
3.1.2 二維之微分方程式.............................................31
3.2 非規則性佈點之探討.............................................39
3.2.1 佈點方式.....................................................39
3.2.2 形狀參數之探討...............................................45
第四章.............................................................53
4.1 不規則區間實例之應用...........................................53
4.1.1 實例一.......................................................53
4.1.2 實例二.......................................................60
4.2 區間切割之探討.................................................66
4.3 區間切割之實例應用.............................................88
第五章.............................................................97
5.1 結論...........................................................97
5.2 展望...........................................................98
參考文獻...........................................................99
附錄..............................................................102
A.奇異值分解（Singular Value Decomposition , SVD） ...............102
B.矩陣之範數（Norm）和條件數（Condition number）..................103

1. Carlson, R. E. and Foley, T. A. ,“The parameter R2 in multiquadric
interpolation”, Comput. Math. Appl., 21, 29-42(1991)
2. Chen, C. S. and Rashed, Y. F. ,“Evaluation of thin plate spline based
particular solution for Helmholtz-type operators for theDRM”, Mechanics
Research Communications, 25, 195-201(1998)
3. Driscoll, T. A. and Fornberg, B., “Interpolation in the of increasinglu flat
radial basis functions”, Comput. Math. Appl., 43, 413-422(2002)
4. Duchon, J. ,“Interpolation des donctions de deux variables suivant le
principe de flexion des plaques minces”, RAIRO Analyse Numeriques, 10,
5-12(1976)
5. Fedoseyev, A. I., Friedman, M. J. and Kansa, E. J., “Improved
multiqudric method for elliptic partial differential equations via PDE
collocation on the boundary”, Compu. Math. Appl., 43, 439-455(2002)
6. Fornberg, B. , Driscoll, T. A. , Wright, G. and Charies, R., “Observations
on the behavior of radial basis function approximation near boundaries”,
Comput. Math. Appl., 43, 473-490 (2002)
7. Franke, R. ,“Scattered data interpolation: tests of some methods”, Math.
Comput., 48, 181-200(1982)
8. Golberg, M. A. ,” Cross-validation for parameter estimation in the BEM”,
Eng. Anal. Bound. Elem., 19 (2): 157-166(1997)
9. Zhou, X., Hon, Y. C. and Li, J.C., ” Overlapping domain decomposition
method by radial basis functions”, Appl. Numer. Math., 44, 241-255
(2003)
10. Hardy, R. L. ,“Multiquadric equations of topography and other irregular
surfaces”, J. Geophys. Res., 176, 1905-1915(1971)
11. Hardy, R. L. ,“Theory and applications of the mutiqudric-biharmonic
method”, Comput. Math. Appl., 19,163-208(1990)
12. Hon, Y. C. and Mao, X. Z., “An sfficient numerical scheme for burgers’
equation”, Appl. Math. Comput., 95, 37-50(1998)
13. Jichun Li , Alexander, H. , Cheng, D. and Chen, C. S. ,”A comparison of
efficiency and error convergence of multiquadric collocation method and
finite element method”, Eng. Anal. Bound. Elem., 27, 251-257(2003)
14. Kansa, E. J. ,“Multiquadric- A scattered data approximation scheme with
applications to computational fluid dynamics: Ι. Surface approximations
and partial differential equations”, Comp. Math. Appl., 19, 127-145(1990)
15. Kansa, E. J. , “Multiquadric- A scattered data approximation scheme with
applications to computational fluid dynamics: Π. Solutions to parabolic,
hyperbolic, and elliptic partial differential equations”, Comp. Math. Appl.,
19, 147-161(1990)
16. Madych, W. R. and Nelson, S. A. ,“Multivariate interpolation and
conditionally positive definite functions, Π”, Math. Comput., 54, 211-230
(1990)
17. Madych, W. R. and Nelson, S. A. ,“Miscellaneous error bounds for
multiquadeic and telated interpolations”, Comput. Math. Applic., 24,
121-138(1992)
18. Mai-Duy, N. and Tran-Cong, T. ,“Numerical solution of differential
equations using multiqudric radial basis function networks”, Neural
Networks, 14, 185-199(2001)
19. Moody, J. and Darken, C. J. ,“Fast learning in networks of locally-tuned
processing units”, Neural Computations, 1, 281-294(1989)
20. Rippa, S. ,”An algorithm for selecting a good value for the parameter c in
radial basis function interpolation”, Adv. Comput. Math., 11, 193-210
(1999)
21. Schaback, R. ,“Error estimates and condition numbers for radial basis
functions”, Adv. Comput. Math., 3, 251-264(1995)
22. Shul, M. V. and Mitel, Y. Y. ,“The multiqudric method of approximation a
topographic surface”, Geodesy Mapp. Photogramm., 16, 13-17 (1974);
translated from Russian for AGU, ACSM ans ASP (1977)
23. Trahan, C. J. and Wyatt, R. E. ,”Radial basis function interpolation in the
quantum trajectory method: optimization of the multiquadric shape
parameter”, J. comput. Phy., 185, 27-49(2003)
24. Wang, J. G. ,”On the optimal shape parameters of redial basis functions
used for 2-D meshless methods”, Comput. Mehtods Appl. Engng., 191,
2611-2630 (2002)
25. Young , D. L. , Tsai, C. C. and Eldho, T. I. ,”Solution of stokes flow using
an iterative DRBEM based on compactly-supported , positive-definite
radial basis function”, Comput. Math. Appl., 43, 607-619(2002)
26. Zurroukat, M., Power, H. and Chen, C. S. ,”A numerical method for heat
transfer problem using collocation and radial basis functions”, Int. J.
Number. Mesh Engng., 42, 1263-78(1998)