### Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation

Received: 28 May 2020     Accepted: 13 July 2020     Published: 13 August 2020
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Abstract

Partial differential equations are very important tools for mathematical modeling in some field like; physics, engineering and applied Mathematics. It’s worth knowing that only few of this equation can be solved analytically and numerical method have been proven to perform exceedingly well in solving even difficult partial differential equations. Finite difference method is a popular numerical method which has been applied extensively to solve partial differential equations. A well known type of this method is the Classical Crank-Nicolson scheme which has been used by different researchers. In this work, we present a modified Crank-Nicolson scheme resulting from the modification of the classical Crank-Nicolson scheme to solve one dimensional parabolic equation. We apply both the Classical Crank-Nicolson scheme and the modified Crank-Nicolson scheme to solve one dimensional parabolic partial differential equation and investigate the results of the different schemes. The computation and results of the two schemes converges faster to the exact solution. It is shown that the modified Crank-Nicolson method is more efficient, reliable and better for solving Parabolic Partial differential equations since it requires less computational work. The method is stable and the convergence is fast when the results of the numerical examples where compared with the results from other existing classical scheme, we found that our method have better accuracy than those methods. Some numerical examples were considered to verify our results.

 Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 6, Issue 3) This article belongs to the Special Issue Computational Mathematics DOI 10.11648/j.ijamtp.20200603.11 Page(s) 35-40 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

Finite Difference Method, Parabolic Equation, Crank-Nicolson Scheme, Modified Crank-Nicolson Method, Stability

References
 [1] C. E Abhulimen and B. J Omowo, Modified Crank-Nicolson Method for solving one dimensional Parabolic equations. IOSR Journal of Mathematics, Volume15, Issue 6 series 3 (2019) pg 60-66. [2] J. Crank and N. Philis, 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] J. Cooper, Introduction to Partial differential Equation with Matlab, Boston, 1958. [4] E. C DuFort and S. P Franel, Conditions in Numerical Treatment of Partial differential equations. Math. Comput. 7 (43) (1953) 135-152. [5] Emenogu George Ndubueze and Oko Nlia, Solutions of parabolic partial differential equations by finite difference methods. Journal of Applied Mathematics 8 (2), 2015, 88-102. [6] S. E Fadugba, O. H Edogbanya, S. C Zelibe, Crank-Nicolson method for solving parabolic partial differential equations. International Journal of Applied Mathematics and Modeling IJA2M, vol 1, (2013) nos 3pp 8-23. [7] A. Fallahzadeh and K. Shakibi, A method to solve Convection-diffusion equation based on homotopy analysis method. Journal of Interpolation and Approximation in scientific computing 1, (2015) pp1-8. [8] Febi Sanjaya and Sudi Mungkasi, A simple but accurate explicit finite difference method for Advection-diffusion equation, Journal of Phy, Conference series 909, (2017). [9] W. Gerald Recktennwald, Finite difference Approximation to the heat equation, Mathematical Method, 8 (34) 2004 pp747-760. [10] H. A Isede, Several examples of Crank-Nicolson method for parabolic partial differential equations. Academia Journal of Scientific Research 1 (4) (2013), 06 3-068. [11] E. Kreyszig, Advanced Engineering Mathematics, USA, John Wiley and Sons, pp 861-865. [12] Neethu Fernandes and Rakhi B, An overview of Crank-Nicolson method to solve parabolic partial differential equations. International Journal of Scientific Research, vol 7, issue 12, 1074-1086. [13] G. D Smith, Numerical Solution of Partial differential equations: Finite difference methods, Oxford Applied Mathematics and Computing science series, Oxford University press, Third edition, 1985. [14] Williams F. Ames, Numerical Methods for Partial differential Equations, Academic press, Inc, Third edition, 1992.
Cite This Article
• APA Style

Omowo Babajide Johnson, Longe Idowu Oluwaseun. (2020). Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation. International Journal of Applied Mathematics and Theoretical Physics, 6(3), 35-40. https://doi.org/10.11648/j.ijamtp.20200603.11

ACS Style

Omowo Babajide Johnson; Longe Idowu Oluwaseun. Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation. Int. J. Appl. Math. Theor. Phys. 2020, 6(3), 35-40. doi: 10.11648/j.ijamtp.20200603.11

AMA Style

Omowo Babajide Johnson, Longe Idowu Oluwaseun. Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation. Int J Appl Math Theor Phys. 2020;6(3):35-40. doi: 10.11648/j.ijamtp.20200603.11

• @article{10.11648/j.ijamtp.20200603.11,
author = {Omowo Babajide Johnson and Longe Idowu Oluwaseun},
title = {Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {6},
number = {3},
pages = {35-40},
doi = {10.11648/j.ijamtp.20200603.11},
url = {https://doi.org/10.11648/j.ijamtp.20200603.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20200603.11},
abstract = {Partial differential equations are very important tools for mathematical modeling in some field like; physics, engineering and applied Mathematics. It’s worth knowing that only few of this equation can be solved analytically and numerical method have been proven to perform exceedingly well in solving even difficult partial differential equations. Finite difference method is a popular numerical method which has been applied extensively to solve partial differential equations. A well known type of this method is the Classical Crank-Nicolson scheme which has been used by different researchers. In this work, we present a modified Crank-Nicolson scheme resulting from the modification of the classical Crank-Nicolson scheme to solve one dimensional parabolic equation. We apply both the Classical Crank-Nicolson scheme and the modified Crank-Nicolson scheme to solve one dimensional parabolic partial differential equation and investigate the results of the different schemes. The computation and results of the two schemes converges faster to the exact solution. It is shown that the modified Crank-Nicolson method is more efficient, reliable and better for solving Parabolic Partial differential equations since it requires less computational work. The method is stable and the convergence is fast when the results of the numerical examples where compared with the results from other existing classical scheme, we found that our method have better accuracy than those methods. Some numerical examples were considered to verify our results.},
year = {2020}
}

• TY  - JOUR
T1  - Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation
AU  - Omowo Babajide Johnson
AU  - Longe Idowu Oluwaseun
Y1  - 2020/08/13
PY  - 2020
N1  - https://doi.org/10.11648/j.ijamtp.20200603.11
DO  - 10.11648/j.ijamtp.20200603.11
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  - 35
EP  - 40
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20200603.11
AB  - Partial differential equations are very important tools for mathematical modeling in some field like; physics, engineering and applied Mathematics. It’s worth knowing that only few of this equation can be solved analytically and numerical method have been proven to perform exceedingly well in solving even difficult partial differential equations. Finite difference method is a popular numerical method which has been applied extensively to solve partial differential equations. A well known type of this method is the Classical Crank-Nicolson scheme which has been used by different researchers. In this work, we present a modified Crank-Nicolson scheme resulting from the modification of the classical Crank-Nicolson scheme to solve one dimensional parabolic equation. We apply both the Classical Crank-Nicolson scheme and the modified Crank-Nicolson scheme to solve one dimensional parabolic partial differential equation and investigate the results of the different schemes. The computation and results of the two schemes converges faster to the exact solution. It is shown that the modified Crank-Nicolson method is more efficient, reliable and better for solving Parabolic Partial differential equations since it requires less computational work. The method is stable and the convergence is fast when the results of the numerical examples where compared with the results from other existing classical scheme, we found that our method have better accuracy than those methods. Some numerical examples were considered to verify our results.
VL  - 6
IS  - 3
ER  -
Author Information
• Department of Mathematics, Nasarawa State University, Keffi, Nigeria

• Department of Statistics, Federal Polytechnic, Ile-Oluji, Nigeria

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