On the Solution of a Boundary Value Problem related to the Heat Transmission

Khanlar R. Mamedov; Volkan Ala

doi:doi:10.11648/j.ajam.20140202.12

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On the Solution of a Boundary Value Problem related to the Heat Transmission

Khanlar R. Mamedov, Volkan Ala

Published in American Journal of Applied Mathematics (Volume 2, Issue 2)

Received: 28 March 2014 Accepted: 18 April 2014 Published: 30 April 2014

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Abstract

In this study, we consider a heat transmission problem which has derivative with respect to the time in boundary condition. Applying the seperation of variables method, we get a Sturm-Liouville equation with discontinuous coefficient and a spectral parameter dependent boundary condition. For this spectral problem, the operator theoretic formula is given, the resolvent operator constructed and the expansion formula with respect to the eigenfunctions obtained. Using the expansion formula, the solution of the heat problem expressed.

Published in	American Journal of Applied Mathematics (Volume 2, Issue 2)
DOI	10.11648/j.ajam.20140202.12
Page(s)	54-59
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 Operator, Resolvent Operator, Expansion Formula

References

[1]	D. S. Cohen, "An Integral Transform Associated with Boundary Conditions Containing an Eigenvalue Parameter," SIAM Journal on Applied Mathematics, Vol. 14, No. 5 (Sep., 1966), 1164-1175
[2]	C. T. Fulton, "Two-Point Boundary Value Problems with Eigenvalue Parameter Contained in the Boundary Conditions," In. Proc. Roy. Soc. Edin., 77 A, (1977), 293-308.
[3]	C. T. Fulton, "Singuler Eigenvalue Problems with Eigenvalue Parameter Contained in the Boundary Conditions," In. Proc. Roy. Soc. Edin., Section A, Vol. 87, No. 1-2, pp. 1-34, 1980/1
[4]	Levitan B. M., Sargsjan I. S., Introduction to Spectral Theory, American Mathematical Society, 1975.
[5]	E. C. Titchmarsh, Eigenfunction Expansions, Oxford, (1962)
[6]	Naimark M. A., Linear Differantial Operators, Frederick Ungar Publishing, 1967.
[7]	V. A. Marchenko, Sturm-Liouville Operators and Applications, AMS Chelsea Publishing, 2011
[8]	Zettl A., Sturm-Liouville Theory, Mathematical Surveys and Monographs 121, American Math. Soc., Providince, RI, 2005.
[9]	J. Wedmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Math. 1258, Springer-Verlag, Berlin, 1987.
[10]	Kh. R. Mamedov and N. Palamut, "On a Direct Problem of Scattering Theory For A Class Of Sturm Liouville Operator With Discontinous Coefficient," Proccedings of the Jangjean Mathematical Society, vol. 12, no.2, pp. 243-251, 2009.
[11]	Kh. R. Mamedov, "Uniqueness of the Solution of the Inverse Problem of Scattering Theory for Sturm-Liouville Operator with a Spectral Parameter in the Boundary Condition, In: Math. Notes, V. 74 No:1, (2003), 136-140.
[12]	Kh. R. Mamedov, "Uniqueness of the Solution of the Inverse Problem of Scattering Theory for Sturm-Liouville Operator with Discontinous Coefficient," In: Proceedings of IMM of NAS Azerbaijan, 24, (2006), 263-272.
[13]	Kh. R. Mamedov, "On An Inverse Scattering Problem For A Discontinuous Sturm-Liouville Equation With A Spectral Parameter in the Boundary Condition," Boundary Value Problems Volume 2010, article d 171967, 17 pages, doi 10.1155/2010/171967
[14]	M. Guseinov, R.T. Pashaev, "On an Inverse Problem for a Second Order Differential Equation," In: Usp. Math. Nauk, V 57 no: 3, (2002), 597-598J.

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Khanlar R. Mamedov, Volkan Ala. (2014). On the Solution of a Boundary Value Problem related to the Heat Transmission. American Journal of Applied Mathematics, 2(2), 54-59. https://doi.org/10.11648/j.ajam.20140202.12

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Khanlar R. Mamedov; Volkan Ala. On the Solution of a Boundary Value Problem related to the Heat Transmission. Am. J. Appl. Math. 2014, 2(2), 54-59. doi: 10.11648/j.ajam.20140202.12

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AMA Style

Khanlar R. Mamedov, Volkan Ala. On the Solution of a Boundary Value Problem related to the Heat Transmission. Am J Appl Math. 2014;2(2):54-59. doi: 10.11648/j.ajam.20140202.12

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@article{10.11648/j.ajam.20140202.12,
  author = {Khanlar R. Mamedov and Volkan Ala},
  title = {On the Solution of a Boundary Value Problem related to the Heat Transmission},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {2},
  pages = {54-59},
  doi = {10.11648/j.ajam.20140202.12},
  url = {https://doi.org/10.11648/j.ajam.20140202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140202.12},
  abstract = {In this study, we consider a heat transmission problem which has derivative with respect to the time in boundary condition. Applying the seperation of variables method, we get a Sturm-Liouville equation with discontinuous coefficient and a spectral parameter dependent boundary condition. For this spectral problem, the operator theoretic formula is given, the resolvent operator constructed and the expansion formula with respect to the eigenfunctions obtained. Using the expansion formula, the solution of the heat problem expressed.},
 year = {2014}
}

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TY  - JOUR
T1  - On the Solution of a Boundary Value Problem related to the Heat Transmission
AU  - Khanlar R. Mamedov
AU  - Volkan Ala
Y1  - 2014/04/30
PY  - 2014
N1  - https://doi.org/10.11648/j.ajam.20140202.12
DO  - 10.11648/j.ajam.20140202.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 54
EP  - 59
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20140202.12
AB  - In this study, we consider a heat transmission problem which has derivative with respect to the time in boundary condition. Applying the seperation of variables method, we get a Sturm-Liouville equation with discontinuous coefficient and a spectral parameter dependent boundary condition. For this spectral problem, the operator theoretic formula is given, the resolvent operator constructed and the expansion formula with respect to the eigenfunctions obtained. Using the expansion formula, the solution of the heat problem expressed.
VL  - 2
IS  - 2
ER  -

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Author Information

Khanlar R. Mamedov

Mathematics Department, Science and Letters Faculty, Mersin University, Mersin, Turkey
Volkan Ala

Mathematics Department, Science and Letters Faculty, Mersin University, Mersin, Turkey

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APA Style

Khanlar R. Mamedov, Volkan Ala. (2014). On the Solution of a Boundary Value Problem related to the Heat Transmission. American Journal of Applied Mathematics, 2(2), 54-59. https://doi.org/10.11648/j.ajam.20140202.12

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ACS Style

Khanlar R. Mamedov; Volkan Ala. On the Solution of a Boundary Value Problem related to the Heat Transmission. Am. J. Appl. Math. 2014, 2(2), 54-59. doi: 10.11648/j.ajam.20140202.12

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AMA Style

Khanlar R. Mamedov, Volkan Ala. On the Solution of a Boundary Value Problem related to the Heat Transmission. Am J Appl Math. 2014;2(2):54-59. doi: 10.11648/j.ajam.20140202.12

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@article{10.11648/j.ajam.20140202.12,
  author = {Khanlar R. Mamedov and Volkan Ala},
  title = {On the Solution of a Boundary Value Problem related to the Heat Transmission},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {2},
  pages = {54-59},
  doi = {10.11648/j.ajam.20140202.12},
  url = {https://doi.org/10.11648/j.ajam.20140202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140202.12},
  abstract = {In this study, we consider a heat transmission problem which has derivative with respect to the time in boundary condition. Applying the seperation of variables method, we get a Sturm-Liouville equation with discontinuous coefficient and a spectral parameter dependent boundary condition. For this spectral problem, the operator theoretic formula is given, the resolvent operator constructed and the expansion formula with respect to the eigenfunctions obtained. Using the expansion formula, the solution of the heat problem expressed.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - On the Solution of a Boundary Value Problem related to the Heat Transmission
AU  - Khanlar R. Mamedov
AU  - Volkan Ala
Y1  - 2014/04/30
PY  - 2014
N1  - https://doi.org/10.11648/j.ajam.20140202.12
DO  - 10.11648/j.ajam.20140202.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 54
EP  - 59
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20140202.12
AB  - In this study, we consider a heat transmission problem which has derivative with respect to the time in boundary condition. Applying the seperation of variables method, we get a Sturm-Liouville equation with discontinuous coefficient and a spectral parameter dependent boundary condition. For this spectral problem, the operator theoretic formula is given, the resolvent operator constructed and the expansion formula with respect to the eigenfunctions obtained. Using the expansion formula, the solution of the heat problem expressed.
VL  - 2
IS  - 2
ER  -

Copy | Download