### Analytical Solution vs. Numerical Solution of Heat Equation Flow Through Rod of Length 8 Units in One Dimension

Received: 25 January 2021     Accepted: 7 June 2021     Published: 2 July 2021
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Abstract

The paper presented the basic treatment of the solution of heat equation in one dimension. Heat is a form of energy in transaction and it flows from one system to another if there is a temperature difference between the systems. Heat flow is the main concern of sciences which seeks to predict the energy transfer which may take place between material bodies as result of temperature difference. Thus, there are three modes of heat transfer, i.e., conduction, radiation and convection. Conduction can be steady state heat conduction, or unsteady state heat conduction. If the system is in steady state, temperature doesn’t vary with time, but if the system in unsteady state temperature may varies with time. However, if the temperature of material is changing with time or if there are heat sources or sinks within the material the situation is more complex. So, rather than to escape all problem, we are targeted to solve one problem of heat equation in one dimension. The treatment was from both the analytical and the numerical view point, so that the reader is afforded the insight that is gained from analytical solution as well as the numerical solution that must often be used in practice. Analytical we used the techniques of separation of variables. It is worthwhile to mention here that, analytical solution is not always possible to obtain; indeed, in many instants they are very cumber some and difficult to use. In that case a numerical technique is more appropriate. Among numerical techniques finite difference schema is used. In both approach we found a solution which agrees up to one decimal place.

 Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 2) DOI 10.11648/j.ijamtp.20210702.12 Page(s) 53-61 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), 2021. Published by Science Publishing Group
Keywords

Analytical Approach, Numerical Approach, Heat Equation

References
 [1] Yehude Pinch over and Jacob Rubinstein, an introduction to partial differential equation, 2005. [2] Steven Chapra Raymond P Canale, numerical methods for engineers, 7th edition, 2015. [3] J. P. Holman, “Heat transfer”, tenth edition, 2010. [4] David Borthwick, ‘introduction to partial differential equation and it’s application’, 1st edition, 2016. [5] Aslak Tveito, Ragnar Winther, ‘introduction to partial differential equation and it’s application: a computational approach’ 2015. [6] Mehta, Nirajkumar & Vipul, Mr & Gondaliya, Bhavesh & Jayesh, Mr & Gundaniya, V. (2013). Applications of Different Numerical Methods in Heat Transfer - A Review. International Journal of Emerging Technology and Advanced Engineering. 3. 363. [7] Edmund Agyeman, Derick Folson, International Journal of Computer Applications (0975 – 8887) Volume 79 – No5, October 2013 [8] Matthew J. Hancock, The 1-D Heat Equation, 2006 [9] Vitoriano Ruas, Numerical Methods for Partial Differential Equations: An Introduction, 2016 [10] Ahmad, Najmuddin & Charan, Shiv. (2018). STUDY OF NUMERICAL ACCURACY OF ONE DIMENSIONAL HEAT EQUATION BY BENDER-SCHMIDT METHOD, CRANK-NICHOLSON DIFFERENCE METHOD AND DU FORT AND FRANKEL METHOD. [11] Walter A. Strauss, partial differential equations, an introductions, 2008. [12] Subani, Norazlina, Faizzuddin Jamaluddin, Muhammad Arif Hannan Mohamed, and Ahmad Danial Hidayatullah Badrolhisam. "Analytical Solution of Homogeneous One-Dimensional Heat Equation with Neumann Boundary Conditions." In Journal of Physics: Conference Series, vol. 1551, no. 1, p. 012002. IOP Publishing, 2020. [13] Lapidus, Leon, and George F. Pinder. Numerical solution of partial differential equations in science and engineering. John Wiley & Sons, 2011. [14] Agyeman, Edmund, and Derick Folson. "Algorithm analysis of numerical solutions to the heat equation." International Journal of Computer Applications 79, no. 5 (2013). [15] NIGATIE, Y. "The finite difference methods for parabolic partial differential equations." Journal of Applied & Computational Mathematics 7, no. 3 (2018): 1-4. [16] Mohamad, A. N., and Sisay Fikadu. "EFFECT AND EFFICIENT APPROACH OF SOLVING HEAT EQUATION BY DIFFERENT NUMERICAL APPROACHES INCLUDING CRANK–NICOLSON SCHMIDT (METHOD) WHICH IS STILL ACTING AS A MAJOR ROLE." International Journal of Advanced Research in Engineering and Applied Sciences 6, no. 7 (2017): 18-41. [17] Makhtoumi, Mehran. "Numerical solutions of heat diffusion equation over one dimensional rod region." arXiv preprint arXiv: 1807.09588 (2018).
Cite This Article
• APA Style

Geleta Kinkino Meyu, Kedir Aliyi Koriche. (2021). Analytical Solution vs. Numerical Solution of Heat Equation Flow Through Rod of Length 8 Units in One Dimension. International Journal of Applied Mathematics and Theoretical Physics, 7(2), 53-61. https://doi.org/10.11648/j.ijamtp.20210702.12

ACS Style

Geleta Kinkino Meyu; Kedir Aliyi Koriche. Analytical Solution vs. Numerical Solution of Heat Equation Flow Through Rod of Length 8 Units in One Dimension. Int. J. Appl. Math. Theor. Phys. 2021, 7(2), 53-61. doi: 10.11648/j.ijamtp.20210702.12

AMA Style

Geleta Kinkino Meyu, Kedir Aliyi Koriche. Analytical Solution vs. Numerical Solution of Heat Equation Flow Through Rod of Length 8 Units in One Dimension. Int J Appl Math Theor Phys. 2021;7(2):53-61. doi: 10.11648/j.ijamtp.20210702.12

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author = {Geleta Kinkino Meyu and Kedir Aliyi Koriche},
title = {Analytical Solution vs. Numerical Solution of Heat Equation Flow Through Rod of Length 8 Units in One Dimension},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {7},
number = {2},
pages = {53-61},
doi = {10.11648/j.ijamtp.20210702.12},
url = {https://doi.org/10.11648/j.ijamtp.20210702.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210702.12},
abstract = {The paper presented the basic treatment of the solution of heat equation in one dimension. Heat is a form of energy in transaction and it flows from one system to another if there is a temperature difference between the systems. Heat flow is the main concern of sciences which seeks to predict the energy transfer which may take place between material bodies as result of temperature difference. Thus, there are three modes of heat transfer, i.e., conduction, radiation and convection. Conduction can be steady state heat conduction, or unsteady state heat conduction. If the system is in steady state, temperature doesn’t vary with time, but if the system in unsteady state temperature may varies with time. However, if the temperature of material is changing with time or if there are heat sources or sinks within the material the situation is more complex. So, rather than to escape all problem, we are targeted to solve one problem of heat equation in one dimension. The treatment was from both the analytical and the numerical view point, so that the reader is afforded the insight that is gained from analytical solution as well as the numerical solution that must often be used in practice. Analytical we used the techniques of separation of variables. It is worthwhile to mention here that, analytical solution is not always possible to obtain; indeed, in many instants they are very cumber some and difficult to use. In that case a numerical technique is more appropriate. Among numerical techniques finite difference schema is used. In both approach we found a solution which agrees up to one decimal place.},
year = {2021}
}
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AB  - The paper presented the basic treatment of the solution of heat equation in one dimension. Heat is a form of energy in transaction and it flows from one system to another if there is a temperature difference between the systems. Heat flow is the main concern of sciences which seeks to predict the energy transfer which may take place between material bodies as result of temperature difference. Thus, there are three modes of heat transfer, i.e., conduction, radiation and convection. Conduction can be steady state heat conduction, or unsteady state heat conduction. If the system is in steady state, temperature doesn’t vary with time, but if the system in unsteady state temperature may varies with time. However, if the temperature of material is changing with time or if there are heat sources or sinks within the material the situation is more complex. So, rather than to escape all problem, we are targeted to solve one problem of heat equation in one dimension. The treatment was from both the analytical and the numerical view point, so that the reader is afforded the insight that is gained from analytical solution as well as the numerical solution that must often be used in practice. Analytical we used the techniques of separation of variables. It is worthwhile to mention here that, analytical solution is not always possible to obtain; indeed, in many instants they are very cumber some and difficult to use. In that case a numerical technique is more appropriate. Among numerical techniques finite difference schema is used. In both approach we found a solution which agrees up to one decimal place.
VL  - 7
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Author Information
• Department of Mathematics, College of Natural and Computational Science, Ambo University, Ambo, Ethiopia

• Department of Mathematics, College of Natural and Computational Science, Ambo University, Ambo, Ethiopia

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