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

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]
Cite This Article
  • APA Style

    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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    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.
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Author Information
  • Department of Calculative Mathematic and Cybernatic, Moscow, Russia

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