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

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.

DOI	10.11648/j.acm.20150405.16
Published in	Applied and Computational Mathematics (Volume 4, Issue 5, October 2015)
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), 2024. 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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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
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  -

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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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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}
}

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