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

Kedir Aliyi Koroche Kedir Aliyi Koroche

doi:doi:10.11648/j.ijamtp.20210702.11

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

Kedir Aliyi Koroche Kedir Aliyi Koroche

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 2)

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

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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), 2024. 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

[1]	Soyoon Bak, Philsu Kim, Xiangfan Piao, and Sunyoung Bu. Numerical solution of advection– Diffusion type equation by modified error correction scheme. Advances in Difference Equations, 2018.
[2]	Ram Jiwari, R. C. Mittal, Kapil K. Sharma. A numerical scheme based on the weighted average differential quadrature method for the numerical solution of Burgers’ equation. Applied Mathematics and Computation 219 (2013) 6680–6691.
[3]	E. N. Aksan, Quadratic B-spline finite element method for the numerical solution of the Burgers’ Equation, Appl. Math. Comput. 174 (2006) 884–896.
[4]	H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev. 43 (1915) 163–170.
[5]	J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., vol. I, Academic Press, New York, 1948. 171-199.
[6]	Ram Tiwari. A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ Equation. Computer Physics Communications 183 (2012) 2413–2423.
[7]	J. M. Burgers, Mathematical example illustrating relations occurring in the theory of turbulent fluid motion, Trans. Roy. Neth. Acad. Sci. Amsterdam 17 (1939) 1–53.
[8]	S. Kutluay, A. Esen, A lumped Galerkin method for solving the Burgers equation, Int. J. Comput. Math. 81 (11) (2004) 1433–1444.
[9]	Wang, Fajie, Chia-Ming Fan, Chuanzeng Zhang, and Ji Lin. "A localized space-time method of fundamental solutions for diffusion and convection-diffusion problems." Adv. Appl. Math. Mech 12, no. 4 (2020): 940-958.
[10]	Sachin S. Wani and Sarita H. Thakar. Crank-Nicolson type method for burgers equation. International Journal of Applied Physics and Mathematics, Vol. 3, No. 5, September 2013.
[11]	Amit Kumar Verma, Mukesh Kumar Rawani, Ravi P. Agarwal. On a seventh order convergent weakly L-stable Newton Cotes formula with application on Burger’s equation. 2019.
[12]	Khater, A. H., Temsah, R. S., and Callebaut, D. K. Numerical solutions for some coupled nonlinear evolution equations by using the spectral collocation method. Mathematical and Computer Modeling, 48 (7-8), 2008, 1237-1253.
[13]	S. Abbasbandy, M. T. Darvishi, A numerical solution of Burgers’ equation by modified Adomian method, Appl. Math. Comput. 163, 2005, 1265–1272.
[14]	J. M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. R. Nether. Acad. Sci. Amsterdam. 43, 1940, 2–12.
[15]	Sirendaoreji, Exact solutions of the two-dimensional Burgers equation, J. Phys. A: Math. Gen. 32, 1999, 6897–6900.
[16]	Gowrisankar, S., and Natesan, S. An efficient robust numerical method for singularly Perturbed Burgers’ equation. Applied Mathematics and Computation 346, 2019, 385-398.
[17]	Reza Abazaria, A. Borhanifar. Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method. Computers and Mathematics with Applications 59, 2010, 2711–2722.
[18]	J. G. Zheng, T. S. Lee, and S. H. Winoto. A piecewise parabolic method for barotropic and non Barotropic two-fluid flows. International Journal of Numerical Methods for Heat & Fluid Flow 18 (6), 2008, 708-729.
[19]	A. Mignone, T. Plewa and G. Bodo. The piecewise parabolic method for multidimensional Relativistic fluid dynamics, 2005.
[20]	Coleslaw, P. and Woodward, P. R., “The numerical simulation of two-dimensional fluid flow with strong shocks”, J. Comput. Phys., Vol. 54, 1984b, 115-73.
[21]	Kedir Aliyi Koroche. Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation. Indian Journal of Advanced Mathematics (IJAM), 1 (1), April 2021.
[22]	Coleslaw, P. and Woodward, P. R. (1984b), “The numerical simulation of two-dimensional fluid flow with strong shocks”, J. Comput. Phys., Vol. 54, pp. 115-73.
[23]	Kedir Aliyi Koroche. Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation. Mathematics and Computer Science. 2020; 5: 76-85.
[24]	M. W. Oday. Stability and Convergence for nonlinear partial differential equations, Master of Science in Mathematics thesis, Boise State University. 2012.
[25]	Rashidinia, J., F. Esfahani, and S. Jamalzadeh. "B-spline collocation approach for the solution of Klein-Gordon equation." 2013: 25-33.
[26]	Shokofeh S. and Rashidinia J. Numerical solution of the hyperbolic telegraph the equation by cubic B-spline collocation method. Applied Mathematics and Computation 281, 2016, 28–38.
[27]	Smith, Gordon D., Gordon D. Smith, and Gordon Dennis Smith. Numerical solution of partial differential equations: finite difference methods. Oxford university press, 1985.
[28]	Thomas, James William. A numerical method for partial differential equations: finite difference methods. Vol. 22. Springer Science & Business Media, 2013.
[29]	SS Xie, S. Heo, S. Kim, G. Woo, and S. Yi, Numerical solution of one-dimensional Burgers equation using reproducing kernel function, Journal of Computational and Applied Mathematics, 214 (2), 2008, 417-434.
[30]	Vijitha Mukundan, Ashish Awasthi. "Numerical Treatment of the Modified Burgers’ Equation via backward Differentiation Formulas of Orders Two and Three", International Journal of Nonlinear Sciences and Numerical Simulation, 2018.
[31]	International Journal of Numerical Methods for Heat & Fluid Flow, Volume 22, Issue 7 (2012-09-22).
[32]	Tahir Nazir, Muhammad Abbas, Muhammad Yaseen. "Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach", Cogent Mathematics, 2017.
[33]	Feng Gao, Chunmei Chi. "Numerical solution of nonlinear Burgers’ equation using high accuracy multi-quadric quasi-interpolation", Applied Mathematics and Computation, 2014.

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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

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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

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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

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@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}
}

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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  -

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Author Information

Kedir Aliyi Koroche Kedir Aliyi Koroche

Department of Mathematics, College of Natural and Computational Sciences, Ambo University, Ambo, Ethiopia

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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

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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

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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

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@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}
}

Copy | Download

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  -

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