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Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions

Received: 26 November 2015     Accepted: 4 December 2015     Published: 25 December 2015
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Abstract

In this paper I present Numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions. Two methods are used to compute the numerical solutions, viz. Finite difference methods and Finite element methods. The finite element methods are implemented by Crank - Nicolson method. The numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions using finite difference methods do not always converge to the exact solutions. It indicates the occurrence of numerical instability in finite difference methods. Finally the numerical solutions obtained by these two methods are compared with the analytic solutions graphically into two and three dimensions.

Published in American Journal of Applied Mathematics (Volume 3, Issue 6)
DOI 10.11648/j.ajam.20150306.20
Page(s) 305-311
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), 2015. Published by Science Publishing Group

Keywords

Initial Condition, Dirichlet Boundary Conditions, Finite Difference Methods, Finite Element Methods, Heat Equation, Instability

References
[1] Alfio Quarteroni, Numerical models for differential problems (2nd edition), Springer - Verlag, Italia, 2014.
[2] Chapra Canal, Numerical methods for engineers (4th edition), The McGraw Hill Companies, 2001.
[3] Erwin Kreyzing, Advanced Engineering Mathematics (9th edition), 2006 John Wiley and Sons, Inc.
[4] G. Evans, J. Blackledge and P. Yardley, Numerical methods for partial differential equations, Springer-Verlag London Limited 2000.
[5] Ioannis P Starroulakis and Stepan A Tersian, Partial Differential equations: An introduction with Mathematica and MAPLE(2nd edition), 2004 by world scientific publishing Co. Plc. Ltd.
[6] J. David Logan, A first course in DEs, 2006 Springer Science + Business Media. Inc.
[7] Lichard L. Burden and J. Douglas Faires, Numerical analysis (9th edition), 2011, 2005, 2001 Brooks/Cole, Cengage Learning, 20 Channel Center Street Boston, MA02210, USA.
[8] Ravi P. Agarwal and Donal O'Regan, Ordinary and Partial DEs, 2006 Springer Science + Business Media, LLC (2009).
[9] Susan C. Brenner, L. Ridgway Scott, The mathematical theory of finite element methods (3rd edition), 2008 Springer Sciences + Business Media, LLC.
[10] V. Dabral, S. Kapoor and S. Dhawan, Numerical Simulation of one dimensional Heat Equation: B-Spline Finite Element Method, V. Dabral et al. / Indian Journal of Computer Science and Engineering (IJCSE), Vol. 2 No. 2 Apr - May 2011.
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  • APA Style

    Benyam Mebrate. (2015). Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions. American Journal of Applied Mathematics, 3(6), 305-311. https://doi.org/10.11648/j.ajam.20150306.20

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    ACS Style

    Benyam Mebrate. Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions. Am. J. Appl. Math. 2015, 3(6), 305-311. doi: 10.11648/j.ajam.20150306.20

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    AMA Style

    Benyam Mebrate. Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions. Am J Appl Math. 2015;3(6):305-311. doi: 10.11648/j.ajam.20150306.20

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  • @article{10.11648/j.ajam.20150306.20,
      author = {Benyam Mebrate},
      title = {Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions},
      journal = {American Journal of Applied Mathematics},
      volume = {3},
      number = {6},
      pages = {305-311},
      doi = {10.11648/j.ajam.20150306.20},
      url = {https://doi.org/10.11648/j.ajam.20150306.20},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150306.20},
      abstract = {In this paper I present Numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions. Two methods are used to compute the numerical solutions, viz. Finite difference methods and Finite element methods. The finite element methods are implemented by Crank - Nicolson method. The numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions using finite difference methods do not always converge to the exact solutions. It indicates the occurrence of numerical instability in finite difference methods. Finally the numerical solutions obtained by these two methods are compared with the analytic solutions graphically into two and three dimensions.},
     year = {2015}
    }
    

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    T1  - Numerical Solution of a One Dimensional Heat Equation with Dirichlet Boundary Conditions
    AU  - Benyam Mebrate
    Y1  - 2015/12/25
    PY  - 2015
    N1  - https://doi.org/10.11648/j.ajam.20150306.20
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    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 305
    EP  - 311
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20150306.20
    AB  - In this paper I present Numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions. Two methods are used to compute the numerical solutions, viz. Finite difference methods and Finite element methods. The finite element methods are implemented by Crank - Nicolson method. The numerical solutions of a one dimensional heat Equation together with initial condition and Dirichlet boundary conditions using finite difference methods do not always converge to the exact solutions. It indicates the occurrence of numerical instability in finite difference methods. Finally the numerical solutions obtained by these two methods are compared with the analytic solutions graphically into two and three dimensions.
    VL  - 3
    IS  - 6
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Author Information
  • School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

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