### Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation

Received: 26 March 2021     Accepted: 21 May 2021     Published: 27 May 2021
Abstract

In this paper, the piecewise parabolic method is presented for solving the one-dimensional advection-diffusion type equation and its application to the burger equation. First, the given solution domain is discretized by using a uniform Discretization grid point. Next by applying the integration in terms of spatial variable, we discretized the given advection-diffusion type equation and converting it into the system of first-order ordinary differential equation in terms of temporarily variable. Next, by using Taylor series expansion we discretized the obtained system of ordinary differential equation and obtain the central finite difference equation. Then using this difference equation, the given advection-diffusion type equation is solved by using the parabolic method at each specified grid point. To validate the applicability of the proposed method, four model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm L2 and maximum absolute error norm L and the local behavior of the solution is captured exactly. Numerical, exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root means square error norm L2 and maximum absolute error norm L presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving advection-diffusion type equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve the advection-diffusion type equation.

 Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 2) DOI 10.11648/j.ijamtp.20210702.11 Page(s) 40-52 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

Advection-diffusion Type Equation, Burger Equation, Piece-wise Parabolic Method, Taylor Series Expansion, Stability, Convergent Analysis, Root Mean Square and Maximum Absolute Error Norm

References
• APA Style

Kedir Aliyi Koroche Kedir Aliyi Koroche. (2021). Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation. International Journal of Applied Mathematics and Theoretical Physics, 7(2), 40-52. https://doi.org/10.11648/j.ijamtp.20210702.11

ACS Style

Kedir Aliyi Koroche Kedir Aliyi Koroche. Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation. Int. J. Appl. Math. Theor. Phys. 2021, 7(2), 40-52. doi: 10.11648/j.ijamtp.20210702.11

AMA Style

Kedir Aliyi Koroche Kedir Aliyi Koroche. Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation. Int J Appl Math Theor Phys. 2021;7(2):40-52. doi: 10.11648/j.ijamtp.20210702.11

• ```@article{10.11648/j.ijamtp.20210702.11,
author = {Kedir Aliyi Koroche Kedir Aliyi Koroche},
title = {Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {7},
number = {2},
pages = {40-52},
doi = {10.11648/j.ijamtp.20210702.11},
url = {https://doi.org/10.11648/j.ijamtp.20210702.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210702.11},
abstract = {In this paper, the piecewise parabolic method is presented for solving the one-dimensional advection-diffusion type equation and its application to the burger equation. First, the given solution domain is discretized by using a uniform Discretization grid point. Next by applying the integration in terms of spatial variable, we discretized the given advection-diffusion type equation and converting it into the system of first-order ordinary differential equation in terms of temporarily variable. Next, by using Taylor series expansion we discretized the obtained system of ordinary differential equation and obtain the central finite difference equation. Then using this difference equation, the given advection-diffusion type equation is solved by using the parabolic method at each specified grid point. To validate the applicability of the proposed method, four model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm L2 and maximum absolute error norm L∞ and the local behavior of the solution is captured exactly. Numerical, exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root means square error norm L2 and maximum absolute error norm L∞ presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving advection-diffusion type equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve the advection-diffusion type equation.},
year = {2021}
}
```
• ```TY  - JOUR
T1  - Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation
AU  - Kedir Aliyi Koroche Kedir Aliyi Koroche
Y1  - 2021/05/27
PY  - 2021
N1  - https://doi.org/10.11648/j.ijamtp.20210702.11
DO  - 10.11648/j.ijamtp.20210702.11
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 40
EP  - 52
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20210702.11
AB  - In this paper, the piecewise parabolic method is presented for solving the one-dimensional advection-diffusion type equation and its application to the burger equation. First, the given solution domain is discretized by using a uniform Discretization grid point. Next by applying the integration in terms of spatial variable, we discretized the given advection-diffusion type equation and converting it into the system of first-order ordinary differential equation in terms of temporarily variable. Next, by using Taylor series expansion we discretized the obtained system of ordinary differential equation and obtain the central finite difference equation. Then using this difference equation, the given advection-diffusion type equation is solved by using the parabolic method at each specified grid point. To validate the applicability of the proposed method, four model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm L2 and maximum absolute error norm L∞ and the local behavior of the solution is captured exactly. Numerical, exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root means square error norm L2 and maximum absolute error norm L∞ presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving advection-diffusion type equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve the advection-diffusion type equation.
VL  - 7
IS  - 2
ER  - ```
Author Information
• Department of Mathematics, College of Natural and Computational Sciences, Ambo University, Ambo, Ethiopia

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