Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method

Virginia Mwelu Kitetu; Thomas Onyango; Jackson Kioko Kwanza; Nicholas Muthama Mutua

doi:doi:10.11648/j.ajam.20130104.14

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Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method

Virginia Mwelu Kitetu, Thomas Onyango, Jackson Kioko Kwanza, Nicholas Muthama Mutua

Published in American Journal of Applied Mathematics (Volume 1, Issue 4)

Received: 7 September 2013 Accepted: Published: 30 October 2013

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Abstract

The present study focuses on determination of temperature distribution in one dimensional bar using Green’s function method. The Green’s Function is obtained using separation of variables, variation formulation principles and Heaviside functions. The Boundary Integral Equation is obtained using the fundamental solution, Divergence theorem, Green Identities, Dirac delta properties and integration by parts. The solution of heat equation given by the Green’s Function and the boundary integral equation has satisfied the uniqueness, regularity and stability of heat equation. The uniqueness, regularity and stability have been proved using smooth properties of class k function, Lyapunov function and Norm. The BEM implementation is performed using FORTRAN 95 software. Solutions to the test problems are presented and time dependence results are in agreement with computed analytical solutions.

Published in	American Journal of Applied Mathematics (Volume 1, Issue 4)
DOI	10.11648/j.ajam.20130104.14
Page(s)	55-70
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

Green’s Function, Heaviside Function, Separation of Variables, Integration by Parts, Lyapunov Function

References

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[2]	Beck, J. V. (1984). Green’s Function Solution for Transient Heat Conduction Problems. International Journal of Heat and Mass Transfer, Vol 27, pg 1244-1253.
[3]	Brebia C. A. (1984). The BEM for Engineers. London: Pentech Press.
[4]	Chang, Y. P. and Tsou, R. C. (1977). Heat Equation in an anisotropic medium Homogenous in Cylindrical Regions-Unsteady State. Journal of Heat Transfer, Vol 99, pg 41-46.
[5]	Cannon K. and John R. (1984). The One Dimensional Heat Equation, 1st ed. London: Menlo Park.
[6]	Cooper M. and Jeffery N. (1998). Introduction to Partial Differential Equations with MATLAB. London: Academic press
[7]	Eduardo, D. G. (2001). Green’s Functions and Numbering System for Transient Heat Conduction. Journal on applied Mathematics, pg 40-57.
[8]	Greenberg, M.D. (1986). Application of Green’s Functions in Science and Engineering: New Jersey, Prentice-Hall, Inc. Editors.
[9]	James, M. H. and Jeffrey, N. D. (1987). Heat Conduction. Mine Ola: Blackwell Scientific Publications, pg 791-895.
[10]	Misawo F. (2011).A solution of One Dimensional Transient Heat Transfer Problem by Boundary Element Method. Cuea.
[11]	Onyango, T.M., Ingham, D.B. and Lensic, D.M. (2008). Restoring Boundary Conditions in Heat Transfer. Journal of engineering mathematics, Vol 62,pg 85-101.
[12]	Ozisik, N. M. (1968). Boundary Value Problems of Heat Conduction. London: Constable and Company Ltd, pg 255.
[13]	Pittis D. and Sissom L. (2004). Heat Transfer, 2nd ed. Newyork: Tata McGraw-Hill, pg 1-3.
[14]	Praprotnik M., Sterk R. and Trobe K. (2002). A new explicit numerical scheme for nonlinear diffusion problems, Parallel numerics: theory and applications, Jozef Stefan Institute and University of Salzburg.
[15]	Stephenson G. and Ardmore, M. P. (1990). Advanced Mathematical Methods for Engineering and Science students .London: Cambridge University Press, pg 192
[16]	Venkataraman N., Peres E. and Delgado I. (2010).Temperature Distribution in Space Mounting Plates with Discrete Heat Generation Source Due to Conductive Heat Transfer. USA: Acta Astronautic, Vol 1, pg 90.
[17]	Vijun L. (2004).An introduction to the BEM and Applications in modeling composite materials. Research paper.
[18]	Weisstein M. and Eric W. (1999). Heat Conduction Equation. Journal of Applied Mathematics, Vol 1, pg 1-3.

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APA Style

Virginia Mwelu Kitetu, Thomas Onyango, Jackson Kioko Kwanza, Nicholas Muthama Mutua. (2013). Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method. American Journal of Applied Mathematics, 1(4), 55-70. https://doi.org/10.11648/j.ajam.20130104.14

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ACS Style

Virginia Mwelu Kitetu; Thomas Onyango; Jackson Kioko Kwanza; Nicholas Muthama Mutua. Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method. Am. J. Appl. Math. 2013, 1(4), 55-70. doi: 10.11648/j.ajam.20130104.14

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AMA Style

Virginia Mwelu Kitetu, Thomas Onyango, Jackson Kioko Kwanza, Nicholas Muthama Mutua. Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method. Am J Appl Math. 2013;1(4):55-70. doi: 10.11648/j.ajam.20130104.14

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@article{10.11648/j.ajam.20130104.14,
  author = {Virginia Mwelu Kitetu and Thomas Onyango and Jackson Kioko Kwanza and Nicholas Muthama Mutua},
  title = {Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method},
  journal = {American Journal of Applied Mathematics},
  volume = {1},
  number = {4},
  pages = {55-70},
  doi = {10.11648/j.ajam.20130104.14},
  url = {https://doi.org/10.11648/j.ajam.20130104.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20130104.14},
  abstract = {The present study focuses on determination of temperature distribution in one dimensional bar using Green’s function method. The Green’s Function is obtained using separation of variables, variation formulation principles and Heaviside functions. The Boundary Integral Equation is obtained using the fundamental solution, Divergence theorem, Green Identities, Dirac delta properties and integration by parts. The solution of heat equation given by the Green’s Function and the boundary integral equation has satisfied the uniqueness, regularity and stability of heat equation. The uniqueness, regularity and stability have been proved using smooth properties of class k function, Lyapunov function and    Norm. The BEM implementation is performed using FORTRAN 95 software. Solutions to the test problems are presented and time dependence results are in agreement with computed analytical solutions.},
 year = {2013}
}

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TY  - JOUR
T1  - Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method
AU  - Virginia Mwelu Kitetu
AU  - Thomas Onyango
AU  - Jackson Kioko Kwanza
AU  - Nicholas Muthama Mutua
Y1  - 2013/10/30
PY  - 2013
N1  - https://doi.org/10.11648/j.ajam.20130104.14
DO  - 10.11648/j.ajam.20130104.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 55
EP  - 70
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20130104.14
AB  - The present study focuses on determination of temperature distribution in one dimensional bar using Green’s function method. The Green’s Function is obtained using separation of variables, variation formulation principles and Heaviside functions. The Boundary Integral Equation is obtained using the fundamental solution, Divergence theorem, Green Identities, Dirac delta properties and integration by parts. The solution of heat equation given by the Green’s Function and the boundary integral equation has satisfied the uniqueness, regularity and stability of heat equation. The uniqueness, regularity and stability have been proved using smooth properties of class k function, Lyapunov function and    Norm. The BEM implementation is performed using FORTRAN 95 software. Solutions to the test problems are presented and time dependence results are in agreement with computed analytical solutions.
VL  - 1
IS  - 4
ER  -

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Author Information

Virginia Mwelu Kitetu

Department of Mathematics and Computer Science, Faculty of Science, the Catholic University of Eastern Africa, Nairobi, Kenya
Thomas Onyango

Department of Mathematics and Computer Science, Faculty of Science, the Catholic University of Eastern Africa, Nairobi, Kenya
Jackson Kioko Kwanza

Department of Pure & Applied Mathematics, Jomo Kenyatta University of Agriculture & Technology, Nairobi, Kenya
Nicholas Muthama Mutua

Department of Mathematics and Informatics, School of Science and Informatics, Taita Taveta University College, Voi, Kenya

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Plain Text BibTeX RIS

APA Style

Virginia Mwelu Kitetu, Thomas Onyango, Jackson Kioko Kwanza, Nicholas Muthama Mutua. (2013). Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method. American Journal of Applied Mathematics, 1(4), 55-70. https://doi.org/10.11648/j.ajam.20130104.14

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ACS Style

Virginia Mwelu Kitetu; Thomas Onyango; Jackson Kioko Kwanza; Nicholas Muthama Mutua. Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method. Am. J. Appl. Math. 2013, 1(4), 55-70. doi: 10.11648/j.ajam.20130104.14

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AMA Style

Virginia Mwelu Kitetu, Thomas Onyango, Jackson Kioko Kwanza, Nicholas Muthama Mutua. Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method. Am J Appl Math. 2013;1(4):55-70. doi: 10.11648/j.ajam.20130104.14

Copy | Download

@article{10.11648/j.ajam.20130104.14,
  author = {Virginia Mwelu Kitetu and Thomas Onyango and Jackson Kioko Kwanza and Nicholas Muthama Mutua},
  title = {Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method},
  journal = {American Journal of Applied Mathematics},
  volume = {1},
  number = {4},
  pages = {55-70},
  doi = {10.11648/j.ajam.20130104.14},
  url = {https://doi.org/10.11648/j.ajam.20130104.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20130104.14},
  abstract = {The present study focuses on determination of temperature distribution in one dimensional bar using Green’s function method. The Green’s Function is obtained using separation of variables, variation formulation principles and Heaviside functions. The Boundary Integral Equation is obtained using the fundamental solution, Divergence theorem, Green Identities, Dirac delta properties and integration by parts. The solution of heat equation given by the Green’s Function and the boundary integral equation has satisfied the uniqueness, regularity and stability of heat equation. The uniqueness, regularity and stability have been proved using smooth properties of class k function, Lyapunov function and    Norm. The BEM implementation is performed using FORTRAN 95 software. Solutions to the test problems are presented and time dependence results are in agreement with computed analytical solutions.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - Determination of One Dimensional Temperature Distribution in Metallic Bar Using Green’S Function Method
AU  - Virginia Mwelu Kitetu
AU  - Thomas Onyango
AU  - Jackson Kioko Kwanza
AU  - Nicholas Muthama Mutua
Y1  - 2013/10/30
PY  - 2013
N1  - https://doi.org/10.11648/j.ajam.20130104.14
DO  - 10.11648/j.ajam.20130104.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 55
EP  - 70
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20130104.14
AB  - The present study focuses on determination of temperature distribution in one dimensional bar using Green’s function method. The Green’s Function is obtained using separation of variables, variation formulation principles and Heaviside functions. The Boundary Integral Equation is obtained using the fundamental solution, Divergence theorem, Green Identities, Dirac delta properties and integration by parts. The solution of heat equation given by the Green’s Function and the boundary integral equation has satisfied the uniqueness, regularity and stability of heat equation. The uniqueness, regularity and stability have been proved using smooth properties of class k function, Lyapunov function and    Norm. The BEM implementation is performed using FORTRAN 95 software. Solutions to the test problems are presented and time dependence results are in agreement with computed analytical solutions.
VL  - 1
IS  - 4
ER  -

Copy | Download