A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values

Khapaeva Tatiana Mikhailovna

doi:doi:10.11648/j.acm.20140304.11

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A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values

Khapaeva Tatiana Mikhailovna

Published in Applied and Computational Mathematics (Volume 3, Issue 4)

Received: 4 June 2014 Accepted: 4 July 2014 Published: 20 July 2014

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Abstract

A variational method for calculation of the eigenfunctions and eigenvalues in the Sturm-Liouville problem with the Neumann boundary values is offered. The method is based on a functional, which is introduced in this work. An appropriate numerical algorithm is developed. Calculations for the three potentials are produced: sin((x-π)2/π), cos(4x) and the high not isosceles triangle. The method is applied to the Sturm-Liouville problem with the Dirichlet boundary values. Some suppositions about the inverse Sturm-Liouville problem are made.

Published in	Applied and Computational Mathematics (Volume 3, Issue 4)
DOI	10.11648/j.acm.20140304.11
Page(s)	117-120
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), 2014. Published by Science Publishing Group

Keywords

Sturm-Liouville Problem, Neumann Boundary Values, Dirichlet Boundary Values, Eigenfunctions, Eigenvalues, Variational Method, Functional, Inversed Sturm-Liuville problem, Algorithm

References

[1]	L. D. Akulenko and S.V. Nesterov “Jeffektivnyj metod issledovanija kolebanij sushhestvenno neodnorodnyh raspredelennyh system”, Prikladnaja matematika i mehanika, vol. 61:3, 1997, pp. 466–478
[2]	A. Kirsch “An Introduction to the Mathematical Theory of Inverse Problems”, 2nd ed, Applied Mathematical Sciences, vol. 120, Springer, 2011
[3]	Borg G. “Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte”, Acta Mathematica, vol. 78, 1946, pp. 1–96
[4]	A. N. Tikhonov and V. Ya. Arsenin, “Methods for Solving Ill-Posed Problems”, Nauka, Moscow, 1986 [in Russian]

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Khapaeva Tatiana Mikhailovna. (2014). A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values. Applied and Computational Mathematics, 3(4), 117-120. https://doi.org/10.11648/j.acm.20140304.11

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ACS Style

Khapaeva Tatiana Mikhailovna. A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values. Appl. Comput. Math. 2014, 3(4), 117-120. doi: 10.11648/j.acm.20140304.11

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AMA Style

Khapaeva Tatiana Mikhailovna. A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values. Appl Comput Math. 2014;3(4):117-120. doi: 10.11648/j.acm.20140304.11

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@article{10.11648/j.acm.20140304.11,
  author = {Khapaeva Tatiana Mikhailovna},
  title = {A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {4},
  pages = {117-120},
  doi = {10.11648/j.acm.20140304.11},
  url = {https://doi.org/10.11648/j.acm.20140304.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.11},
  abstract = {A variational method for calculation of the eigenfunctions and eigenvalues in the Sturm-Liouville problem with the Neumann boundary values is offered. The method is based on a functional, which is introduced in this work. An appropriate numerical algorithm is developed. Calculations for the three potentials are produced: sin((x-π)2/π), cos(4x) and the high not isosceles triangle. The method is applied to the Sturm-Liouville problem with the Dirichlet boundary values. Some suppositions about the inverse Sturm-Liouville problem are made.},
 year = {2014}
}

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TY  - JOUR
T1  - A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values
AU  - Khapaeva Tatiana Mikhailovna
Y1  - 2014/07/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140304.11
DO  - 10.11648/j.acm.20140304.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 117
EP  - 120
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140304.11
AB  - A variational method for calculation of the eigenfunctions and eigenvalues in the Sturm-Liouville problem with the Neumann boundary values is offered. The method is based on a functional, which is introduced in this work. An appropriate numerical algorithm is developed. Calculations for the three potentials are produced: sin((x-π)2/π), cos(4x) and the high not isosceles triangle. The method is applied to the Sturm-Liouville problem with the Dirichlet boundary values. Some suppositions about the inverse Sturm-Liouville problem are made.
VL  - 3
IS  - 4
ER  -

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Author Information

Khapaeva Tatiana Mikhailovna

Department of Calculative Mathematic and Cybernatic, Moscow, Russia

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Khapaeva Tatiana Mikhailovna. (2014). A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values. Applied and Computational Mathematics, 3(4), 117-120. https://doi.org/10.11648/j.acm.20140304.11

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ACS Style

Khapaeva Tatiana Mikhailovna. A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values. Appl. Comput. Math. 2014, 3(4), 117-120. doi: 10.11648/j.acm.20140304.11

Copy | Download

AMA Style

Khapaeva Tatiana Mikhailovna. A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values. Appl Comput Math. 2014;3(4):117-120. doi: 10.11648/j.acm.20140304.11

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@article{10.11648/j.acm.20140304.11,
  author = {Khapaeva Tatiana Mikhailovna},
  title = {A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {4},
  pages = {117-120},
  doi = {10.11648/j.acm.20140304.11},
  url = {https://doi.org/10.11648/j.acm.20140304.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.11},
  abstract = {A variational method for calculation of the eigenfunctions and eigenvalues in the Sturm-Liouville problem with the Neumann boundary values is offered. The method is based on a functional, which is introduced in this work. An appropriate numerical algorithm is developed. Calculations for the three potentials are produced: sin((x-π)2/π), cos(4x) and the high not isosceles triangle. The method is applied to the Sturm-Liouville problem with the Dirichlet boundary values. Some suppositions about the inverse Sturm-Liouville problem are made.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - A Variational Method in the Sturm-Liouville Problem with the Neumann and Dirichlet Boundary Values
AU  - Khapaeva Tatiana Mikhailovna
Y1  - 2014/07/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140304.11
DO  - 10.11648/j.acm.20140304.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 117
EP  - 120
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140304.11
AB  - A variational method for calculation of the eigenfunctions and eigenvalues in the Sturm-Liouville problem with the Neumann boundary values is offered. The method is based on a functional, which is introduced in this work. An appropriate numerical algorithm is developed. Calculations for the three potentials are produced: sin((x-π)2/π), cos(4x) and the high not isosceles triangle. The method is applied to the Sturm-Liouville problem with the Dirichlet boundary values. Some suppositions about the inverse Sturm-Liouville problem are made.
VL  - 3
IS  - 4
ER  -

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