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Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation

Received: 10 March 2020     Accepted: 7 April 2020     Published: 12 October 2020
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Abstract

In this paper, present solution of one-dimensional linear parabolic differential equation by using Forward difference, backward difference, and Crank Nicholson method. First, the solution domain is discretized using the uniform mesh for step length and time step. Then applying the proposed method, we discretize the linear parabolic equation at each grid point and then rearranging the obtained discretization scheme we obtain the system of equation generated with tri-diagonal coefficient matrix. Now applying inverse matrixes method and writing MATLAB code for this inverse matrixes method we obtain the solution of one-dimensional linear parabolic differential equation. The stability of each scheme analyses by using Von-Neumann stability analysis technique. To validate the applicability of the proposed method, two model example are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (E) and Root mean error (E2). Also, condition number (K(A)) and Order of convergence are calculated. The stability of this Three class of numerical method is also guaranteed and also, the comparability of the stability of these three methods is presented by using the graphical and tabular form. The proposed method is validated via the same numerical test example. The present method approximate exact solution very well.

Published in Mathematics and Computer Science (Volume 5, Issue 4)
DOI 10.11648/j.mcs.20200504.12
Page(s) 76-85
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), 2020. Published by Science Publishing Group

Keywords

Linear Parabolic Equation, Implicit Crank Nicholson Method, Root Mean Square Error, Condition Number, Order of Convergence

References
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[3] Cheninguael, A. (2014, March). Numerical Method for the Heat Equation with Dirichlet and Neumann Conditions. In Proceedings of the International Multiconference of Engineering and computer scientists (vol. 1).
[4] Dabral, V., Kapoor, S. and Dhawan, S. (Apr-May 2011). Numerical Simulation of one dimensional Heat Equation: B-Spline Finite Element Method, Indian Jour nal of Computer Science and Engineering. 2 (2), 222-235.
[5] Fasshauer, G. (2007). Meshfree Application Method with Matlab. Interdisciplinary Mathematical Sciences.
[6] Franke, R. (1982). Scattered data interpolation: tests of some methods. Mathematics of computation, 38 (157): 181-200.
[7] Francis B. Hildebrand (1987). Introduction to numerical analysis, Second-edition, Dover Publications-Inc., Canada.
[8] Hooshmandasl M. R Haidari, M., and MaalekGhaini F. M. (2012). Numerical solution of one dimensional heat equation by using the Chebyshev Wavelets method. JACM an open-access journal, 1 (6), 1-7.
[9] Kalyanil, P., and Rao (2013). Numerical solution of heat equation through double Interpolation. IOSR Journal. of math. 6 (6): 58-62.
[10] Li, J. R., and Greengard, L. (2007). On the numerical solution of the heat equation I: Fast solvers in free space. Journal of Computational Physics, 226 (2): 1891-1901.
[11] Muluneh Dingeta, Gemechis File and Tesfaye Aga (2018) Numerical Solution of Second Numerical Solution of Second-Order One Dimensional Linear Hyperbolic Telegraph Equation Ethiop. J. Educ. & Sc. Vol. 14 No 1.
[12] Rashidinia J. Esfahani F. and Jamalzadeh S., (2013), B-spline Collocation Approach for Solution of Klein-Gordon Equation. International Journal of Mathematical Modeling and Computations 3, 25-33.
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[15] Shokofeh S. and Rashidinia J. (2016), Numerical solution of the hyperbolic telegraph the equation by cubic B-spline collocation method. Applied Mathematics and Computation 281, 28–38.
[16] Sastry, S. S. (2006). Introductory method of numerical analysis, Fourth-edition, Asoke Ghash, prentice Hall of India.
[17] Schiesser, W. E., and Griffiths, G. W. (2009). A compendium of partial different tial equation models: method of lines analysis with Matlab. Cambridge University Press.
[18] Tatari, M. and Dehghan, M., (2010). A method for solving partial differential equations via radial basis functions: Application to the heat equation Engineering Analysis with Boundary element 34 (3), 206-212.
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[20] Hikmet¸ ag˘lar, Mehmet O. zer, Nazan¸ ag˘lar (2008) The numerical solution of the one-dimensional heat equation by using third degree B-spline functions.
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  • APA Style

    Kedir Aliyi Koroche. (2020). Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation. Mathematics and Computer Science, 5(4), 76-85. https://doi.org/10.11648/j.mcs.20200504.12

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    ACS Style

    Kedir Aliyi Koroche. Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation. Math. Comput. Sci. 2020, 5(4), 76-85. doi: 10.11648/j.mcs.20200504.12

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    AMA Style

    Kedir Aliyi Koroche. Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation. Math Comput Sci. 2020;5(4):76-85. doi: 10.11648/j.mcs.20200504.12

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  • @article{10.11648/j.mcs.20200504.12,
      author = {Kedir Aliyi Koroche},
      title = {Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation},
      journal = {Mathematics and Computer Science},
      volume = {5},
      number = {4},
      pages = {76-85},
      doi = {10.11648/j.mcs.20200504.12},
      url = {https://doi.org/10.11648/j.mcs.20200504.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20200504.12},
      abstract = {In this paper, present solution of one-dimensional linear parabolic differential equation by using Forward difference, backward difference, and Crank Nicholson method. First, the solution domain is discretized using the uniform mesh for step length and time step. Then applying the proposed method, we discretize the linear parabolic equation at each grid point and then rearranging the obtained discretization scheme we obtain the system of equation generated with tri-diagonal coefficient matrix. Now applying inverse matrixes method and writing MATLAB code for this inverse matrixes method we obtain the solution of one-dimensional linear parabolic differential equation. The stability of each scheme analyses by using Von-Neumann stability analysis technique. To validate the applicability of the proposed method, two model example are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (E∞) and Root mean error (E2). Also, condition number (K(A)) and Order of convergence are calculated. The stability of this Three class of numerical method is also guaranteed and also, the comparability of the stability of these three methods is presented by using the graphical and tabular form. The proposed method is validated via the same numerical test example. The present method approximate exact solution very well.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation
    AU  - Kedir Aliyi Koroche
    Y1  - 2020/10/12
    PY  - 2020
    N1  - https://doi.org/10.11648/j.mcs.20200504.12
    DO  - 10.11648/j.mcs.20200504.12
    T2  - Mathematics and Computer Science
    JF  - Mathematics and Computer Science
    JO  - Mathematics and Computer Science
    SP  - 76
    EP  - 85
    PB  - Science Publishing Group
    SN  - 2575-6028
    UR  - https://doi.org/10.11648/j.mcs.20200504.12
    AB  - In this paper, present solution of one-dimensional linear parabolic differential equation by using Forward difference, backward difference, and Crank Nicholson method. First, the solution domain is discretized using the uniform mesh for step length and time step. Then applying the proposed method, we discretize the linear parabolic equation at each grid point and then rearranging the obtained discretization scheme we obtain the system of equation generated with tri-diagonal coefficient matrix. Now applying inverse matrixes method and writing MATLAB code for this inverse matrixes method we obtain the solution of one-dimensional linear parabolic differential equation. The stability of each scheme analyses by using Von-Neumann stability analysis technique. To validate the applicability of the proposed method, two model example are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (E∞) and Root mean error (E2). Also, condition number (K(A)) and Order of convergence are calculated. The stability of this Three class of numerical method is also guaranteed and also, the comparability of the stability of these three methods is presented by using the graphical and tabular form. The proposed method is validated via the same numerical test example. The present method approximate exact solution very well.
    VL  - 5
    IS  - 4
    ER  - 

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Author Information
  • Department of Mathematics, College of Natural Sciences, Ambo University, Ambo, Ethiopia

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