Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem

Dong Yun Shen; Yong Huang

doi:doi:10.11648/j.acm.20150405.16

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Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem

Dong Yun Shen, Yong Huang

Published in Applied and Computational Mathematics (Volume 4, Issue 5)

Received: 31 August 2015 Accepted: 11 September 2015 Published: 22 September 2015

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Abstract

This paper discusses the eigenvalue problem of second-order Sturm-Liouville equation. We transform the governing differential equation to the Fredholm-Volterra integral equation with appropriate end supports. By expanding the unknown function into the shifted Chebyshev polynomials, we directly get the corresponding polynomial characteristic equations, where the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples of estimating eigenvalues are given. By comparison with the exact results in open literatures, the correctness and effectiveness of the present approach are verified.

Published in	Applied and Computational Mathematics (Volume 4, Issue 5)
DOI	10.11648/j.acm.20150405.16
Page(s)	369-373
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Sturm-Liouville Problem, Eigenvalues, Fredholm-Volterra Integral Equation, Chebyshev Polynomials

References

[1]	MA Al-Gwaiz, Sturm–Liouville Theory and its Applications, Springer-Verlag, London, 2008.
[2]	Veerle Ledoux, Study of special algorithms for solving Sturm–Liouville and Schrödinger equations. Ph.D Thesis, Department of Applied Mathematics and Computer Science, Ghent University, 2007.
[3]	Alan L Andrew, Correction of finite difference eigenvalues of periodic Sturm–Liouville problems, J. Austral. Math. Soc. Ser. B 30 (1989) 460-469.
[4]	Paolo Ghelardoni, Approximations of Sturm--Liouville eigenvalues using boundary value methods, Appl. Numer. Math. 23 (1997) 311--325.
[5]	DJ Condon, Corrected finite difference eigenvalues of periodic Sturm–Liouville problems, Appl. Numer. Math. 30 (1999) 393–401.
[6]	İbrahim Çelik, Approximate computation of eigenvalues with Chebyshev collocation method, Appl. Math. Comput. 168 (2005) 125–134.
[7]	İbrahim Çelik, Guzin Gokmen, Approximate solution of periodic Sturm–Liouville problems with Chebyshev collocation method, Appl. Math. Comput. 170 (2005) 285–295.
[8]	Quan Yuan, Zhiqing He, Huinan Leng, An improvement for Chebyshev collocation method in solving certain Sturm–Liouville problems, Appl. Math. Comput. 195 (2008) 440–447.
[9]	Lan Chen, He-Ping Ma, Approximate solution of the Sturm–Liouville problems with Legendre–Galerkin–Chebyshev collocation method, Appl. Math. Comput. 206 (2008) 748–754.
[10]	Xuecang Zhang, Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm–Liouville problems, Appl. Math. Comput. 217 (2010) 2266-2276.
[11]	Saeid Abbasbandy, A new application of the homotopy analysis method: Solving the Sturm–Liouville problems, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 112–126.
[12]	Mohamed El-gamel, Mahmoud Abd El-hady, Two very accurate and efficient methods for computing eigenvalues of Sturm–Liouville problems, Appl. Math. Modell. 37 (2013) 5039–5046.
[13]	Alan L Andrew, Twenty years of asymptotic correction for eigenvalue computation, ANZIAM J. 42 (2000) C96–C116.

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APA Style

Dong Yun Shen, Yong Huang. (2015). Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem. Applied and Computational Mathematics, 4(5), 369-373. https://doi.org/10.11648/j.acm.20150405.16

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ACS Style

Dong Yun Shen; Yong Huang. Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem. Appl. Comput. Math. 2015, 4(5), 369-373. doi: 10.11648/j.acm.20150405.16

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AMA Style

Dong Yun Shen, Yong Huang. Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem. Appl Comput Math. 2015;4(5):369-373. doi: 10.11648/j.acm.20150405.16

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@article{10.11648/j.acm.20150405.16,
  author = {Dong Yun Shen and Yong Huang},
  title = {Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {5},
  pages = {369-373},
  doi = {10.11648/j.acm.20150405.16},
  url = {https://doi.org/10.11648/j.acm.20150405.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150405.16},
  abstract = {This paper discusses the eigenvalue problem of second-order Sturm-Liouville equation. We transform the governing differential equation to the Fredholm-Volterra integral equation with appropriate end supports. By expanding the unknown function into the shifted Chebyshev polynomials, we directly get the corresponding polynomial characteristic equations, where the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples of estimating eigenvalues are given. By comparison with the exact results in open literatures, the correctness and effectiveness of the present approach are verified.},
 year = {2015}
}

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TY  - JOUR
T1  - Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem
AU  - Dong Yun Shen
AU  - Yong Huang
Y1  - 2015/09/22
PY  - 2015
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DO  - 10.11648/j.acm.20150405.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
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EP  - 373
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150405.16
AB  - This paper discusses the eigenvalue problem of second-order Sturm-Liouville equation. We transform the governing differential equation to the Fredholm-Volterra integral equation with appropriate end supports. By expanding the unknown function into the shifted Chebyshev polynomials, we directly get the corresponding polynomial characteristic equations, where the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples of estimating eigenvalues are given. By comparison with the exact results in open literatures, the correctness and effectiveness of the present approach are verified.
VL  - 4
IS  - 5
ER  -

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Author Information

Dong Yun Shen

Department of Mathematics, Foshan University, Foshan, Guangdong, China
Yong Huang

Department of Mathematics, Foshan University, Foshan, Guangdong, China

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Dong Yun Shen, Yong Huang. (2015). Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem. Applied and Computational Mathematics, 4(5), 369-373. https://doi.org/10.11648/j.acm.20150405.16

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ACS Style

Dong Yun Shen; Yong Huang. Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem. Appl. Comput. Math. 2015, 4(5), 369-373. doi: 10.11648/j.acm.20150405.16

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AMA Style

Dong Yun Shen, Yong Huang. Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem. Appl Comput Math. 2015;4(5):369-373. doi: 10.11648/j.acm.20150405.16

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@article{10.11648/j.acm.20150405.16,
  author = {Dong Yun Shen and Yong Huang},
  title = {Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {5},
  pages = {369-373},
  doi = {10.11648/j.acm.20150405.16},
  url = {https://doi.org/10.11648/j.acm.20150405.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150405.16},
  abstract = {This paper discusses the eigenvalue problem of second-order Sturm-Liouville equation. We transform the governing differential equation to the Fredholm-Volterra integral equation with appropriate end supports. By expanding the unknown function into the shifted Chebyshev polynomials, we directly get the corresponding polynomial characteristic equations, where the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples of estimating eigenvalues are given. By comparison with the exact results in open literatures, the correctness and effectiveness of the present approach are verified.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem
AU  - Dong Yun Shen
AU  - Yong Huang
Y1  - 2015/09/22
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150405.16
DO  - 10.11648/j.acm.20150405.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 369
EP  - 373
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150405.16
AB  - This paper discusses the eigenvalue problem of second-order Sturm-Liouville equation. We transform the governing differential equation to the Fredholm-Volterra integral equation with appropriate end supports. By expanding the unknown function into the shifted Chebyshev polynomials, we directly get the corresponding polynomial characteristic equations, where the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples of estimating eigenvalues are given. By comparison with the exact results in open literatures, the correctness and effectiveness of the present approach are verified.
VL  - 4
IS  - 5
ER  -

Copy | Download