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. |
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Copyright © The Author(s), 2020. Published by Science Publishing Group |
Linear Parabolic Equation, Implicit Crank Nicholson Method, Root Mean Square Error, Condition Number, Order of Convergence
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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
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
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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Download@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}
}
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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