On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations

Omowo Babajide Johnson; Longe Idowu Oluwaseun; Osakwe Charles Nnamdi

doi:doi:10.11648/j.ijssam.20230801.11

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On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations

Omowo Babajide Johnson, Longe Idowu Oluwaseun, Osakwe Charles Nnamdi

Published in International Journal of Systems Science and Applied Mathematics (Volume 8, Issue 1)

Received: 8 October 2022 Accepted: 28 November 2022 Published: 14 February 2023

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Abstract

This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 8, Issue 1)
DOI	10.11648/j.ijssam.20230801.11
Page(s)	1-6
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

Finite Difference Schemes, Stability, Von Newmann Method, Accuracy, Heat Equation

References

[1]	Abhulimen C. E and Omowo B. J Modified Crank-Nicolson Method for Solving One Dimensional Parabolic Equation, International Journal of Scientific Research, Volume 15, issue 6 series 3, (2019) pp 60-66.
[2]	Crank J and Philis N. A practical method for numerical evaluation of solution of partial differential equation of heat conduction type. Proc. Camb. Phil. Soc. 1 (1996), 50-57.
[3]	Omowo B. J and Abhulimen C. E on the stability of Modified Crank-Nicolson method for Parabolic Partial differential equations. International Journal of Mathematical sciences and Optimization: Theory and Application. Vol 6, No. 2 (2021) pp 862-873.
[4]	Recktenwald G. W Finite difference approximation to the heat equation. http://www.nada.kth.se/ijalap/unmme/FD
[5]	Karatay. I and Bayramoglu S. A A new difference scheme for time fractional heat equations based on the Crank-Nicolson method. Frac. Calc. Appl. Anal. 16 (4), 892-910 (2013).
[6]	Aswin V. S et al: A comparative study of numerical schemes for convection-diffusion equation. Procedia Eng. 127, 621-627 (2015).
[7]	Azda T. M. A. K and Andallah I. S: Stability analysis of finite difference schemes for advection diffusion equation. Banglasdesh. J. Sci Res. 29 (2) 143-151 (2016).
[8]	Mebrate B: Numerical solution of one dimensional heat equation with Dirichlet boundary condition. Am. J. Appl. Math 3 (6), 305-311 (2015).
[9]	Olusegun O. A, Hoe, Y. S, Ogunbode E. B Finite difference Approximation to Heat Equation via C. Journal of Applied Sciences and Sustainability 3, 188-200, (2017).
[10]	Williams F. Ames, Numerical Methods for Partial differential Equations, Academic press, Inc, Third Edition, 1992.
[11]	Smith G. D: Numerical Solution of Partial Differential Equation: Finite Difference Methods. Clarendon Press, Third Edition, Oxford (1985).
[12]	Grewal B. S: Higher Engineering Mathematics, Khanna Publisher, Forty-second edition (2012).
[13]	Adak, M. Mandal N. R: Numerical and experimental study of mitigation of welding distortion. Appl. Math. Model, 34, 146-158 (2018).
[14]	Adak M: Comparison of Explicit and Implicit Finite difference schemes on diffusion equation. http://doi.org/10.1007/978-981-153615-115.

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APA Style

Omowo Babajide Johnson, Longe Idowu Oluwaseun, Osakwe Charles Nnamdi. (2023). On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations. International Journal of Systems Science and Applied Mathematics, 8(1), 1-6. https://doi.org/10.11648/j.ijssam.20230801.11

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ACS Style

Omowo Babajide Johnson; Longe Idowu Oluwaseun; Osakwe Charles Nnamdi. On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations. Int. J. Syst. Sci. Appl. Math. 2023, 8(1), 1-6. doi: 10.11648/j.ijssam.20230801.11

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AMA Style

Omowo Babajide Johnson, Longe Idowu Oluwaseun, Osakwe Charles Nnamdi. On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations. Int J Syst Sci Appl Math. 2023;8(1):1-6. doi: 10.11648/j.ijssam.20230801.11

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@article{10.11648/j.ijssam.20230801.11,
  author = {Omowo Babajide Johnson and Longe Idowu Oluwaseun and Osakwe Charles Nnamdi},
  title = {On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {8},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ijssam.20230801.11},
  url = {https://doi.org/10.11648/j.ijssam.20230801.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20230801.11},
  abstract = {This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.},
 year = {2023}
}

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TY  - JOUR
T1  - On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations
AU  - Omowo Babajide Johnson
AU  - Longe Idowu Oluwaseun
AU  - Osakwe Charles Nnamdi
Y1  - 2023/02/14
PY  - 2023
N1  - https://doi.org/10.11648/j.ijssam.20230801.11
DO  - 10.11648/j.ijssam.20230801.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20230801.11
AB  - This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.
VL  - 8
IS  - 1
ER  -

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Author Information

Omowo Babajide Johnson

Department of Statistics, Federal Polytechnic, Ile-Oluji, Nigeria
Longe Idowu Oluwaseun

Department of Statistics, Federal Polytechnic, Ile-Oluji, Nigeria
Osakwe Charles Nnamdi

Department of Mathematics, Faculty of Natural and Applied Science, Nasarawa State University, Keffi, Nigeria

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APA Style

Omowo Babajide Johnson, Longe Idowu Oluwaseun, Osakwe Charles Nnamdi. (2023). On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations. International Journal of Systems Science and Applied Mathematics, 8(1), 1-6. https://doi.org/10.11648/j.ijssam.20230801.11

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ACS Style

Omowo Babajide Johnson; Longe Idowu Oluwaseun; Osakwe Charles Nnamdi. On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations. Int. J. Syst. Sci. Appl. Math. 2023, 8(1), 1-6. doi: 10.11648/j.ijssam.20230801.11

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AMA Style

Omowo Babajide Johnson, Longe Idowu Oluwaseun, Osakwe Charles Nnamdi. On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations. Int J Syst Sci Appl Math. 2023;8(1):1-6. doi: 10.11648/j.ijssam.20230801.11

Copy | Download

@article{10.11648/j.ijssam.20230801.11,
  author = {Omowo Babajide Johnson and Longe Idowu Oluwaseun and Osakwe Charles Nnamdi},
  title = {On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {8},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ijssam.20230801.11},
  url = {https://doi.org/10.11648/j.ijssam.20230801.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20230801.11},
  abstract = {This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations
AU  - Omowo Babajide Johnson
AU  - Longe Idowu Oluwaseun
AU  - Osakwe Charles Nnamdi
Y1  - 2023/02/14
PY  - 2023
N1  - https://doi.org/10.11648/j.ijssam.20230801.11
DO  - 10.11648/j.ijssam.20230801.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20230801.11
AB  - This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.
VL  - 8
IS  - 1
ER  -

Copy | Download