[1] Huang Bin, Hu Weiqun,. Finite element method of the eigenvaluesof Sturm-Liouville’s problem [J]. Journal of Southeast University (English Edition), 2003, 19 (4): 437-442. [doi:10.3969/j.issn.1003-7985.2003.04.028]

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Finite element method of the eigenvaluesof Sturm-Liouville’s problem()

Sturm-Liouville问题特征值的有限元方法

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

19

Issue:

2003 4

Page:

437-442

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2003-12-30

Info

Title:

Finite element method of the eigenvaluesof Sturm-Liouville’s problem

Sturm-Liouville问题特征值的有限元方法

Author(s):

Huang Bin¹;  Hu Weiqun²

¹Department of Mechatronic Engineering, Jinling Institute of Technology, Nanjing 210038, China
²Information College, Nanjing Forestry University, Nanjing 210037, China

黄滨¹;  胡卫群²

¹金陵科技学院机电工程系, 南京 210038; ²南京林业大学信息学院, 南京 210037

Keywords:

Sturm-Liouville’s problem;  eigenvalue;  eigenfunction;  finite element method

Sturm-Liouville问题;  特征值;  特征函数;  有限元方法

PACS:

O175.1

DOI:

10.3969/j.issn.1003-7985.2003.04.028

Abstract:

This paper considers the finite element method of the approximate value of eigenvalues of Sturm-Liouville’s problem. The proof of our main result is based on the variational method. Linear interpolating functions are made by interpolation method, the problem of the approximate value of eigenvalues becomes the calculation of eigenvlaues of a matrix. Then the finite element method of the approximate value of the eigenvalues is obtained, and accuracy of(n-1)-th approximate value is estimated by n-th approximate value. When n is increased, the accuracy of eigenvalue λ_k is increased. When n is appropriately selected, the accuracy of λ_k we need is obtained. This finite element method is significant both in applications and in theory.

本文构建了计算Sturm-Liouville问题特征值的有限元方法.主要结果的证明运用了变分法.通过构造适当的线性插值函数, 将微分方程特征值的近似计算问题离散化为矩阵特征值计算问题.从而获得了微分方程特征值的近似值的有限元方法, 而且可以用第n次近似值来估计第n-1次的近似值的精确度. 随着n的增大, 特征值λ_k的精确度逐步提高, 只要适当选取n, 就可以求得所要求精确度的特征值的近似值, 这个算法具有广泛的实用价值和理论价值.

References:

[1] Hile G N, Protter M H. Inequalities for eigenvalue of the Laplacian [J]. Indiana Univ Math J, 1980, 29(4): 523-538.
[2] Hile G N, Yeh R Z. Inequalities for eigenvalue of the biharmonic operator [J]. Pacific J Math, 1984, 112(1): 115-133.
[3] Chen Z C, Qian C L. Estimates for discrete spectrum of Laplacian operator with any order [J]. J China Univ Sci Tech, 1990, 20(3):259-265.
[4] Protter M H. Can one hear the shape of a drum?[J]. SIAM Rev, 1987, 29(2):185-197.
[5] Zhen W G, Qian C L. Estimates for eigenvalue of Sturm-Liouville’s problem [J]. J Math Tech, 1992, 8(1):28-32.(in Chinese)
[6] Huang Bin. One computational method of the eigenvalues of the horizontal across vibration problem of beam [J]. Journal of Southeast University(English Edition), 2002, 18(4):277-282.

Memo

Memo:

Biography: Huang Bin(1958—), male, lecturer, binhuang@public1.ptt.js.cn.

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