A Comparison of Implicit and Modified Implicit Finite Difference Schemes for Solving Parabolic Equations
Asian Research Journal of Mathematics,
Page 1-10
DOI:
10.9734/arjom/2022/v18i1030413
Abstract
This paper presents the comparison of implicit scheme and modied implicit scheme for solving parabolic partial differential equations, the modied implicit scheme is compared with the implicit scheme using its stability, local truncation error, derivation and numerical examples. Following this, it was discovered that the modied implicit scheme can be used as an alternative scheme to the implicit scheme for solving problems on parabolic partial differential equations.
Keywords:
- Implicit scheme
- stability
- local truncation error
- modied implicit scheme
- parabolic equations
How to Cite
Johnson, O. B., Oluwaseun, L. I., Nnamdi, O. C., Oghale, A. E., & Chukwuma, E. M. (2022). A Comparison of Implicit and Modified Implicit Finite Difference Schemes for Solving Parabolic Equations. Asian Research Journal of Mathematics, 18(10), 1-10. https://doi.org/10.9734/arjom/2022/v18i1030413
References
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Parabolic Partial differential equations. International Journal of Mathematical Sciences and
Optimization: Theory and Application. 2021;6(2):862-873.
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Nicolson scheme for one dimensional parabolic partial differential equation. International
journal of Applied Mathematics and Theoretical Physics. 2020;6(3):35-40.
Omowo BJ, et al. On the Convergence and Stability of Finite Difference Method for Parabolic
Partial Differential Equations. Journal of Advances in Mathematics and Computer Science.
;36 (10):58-67.
Febi Sanjaya, Sudi Mungkasi. A simple but accurate explicit finite difference method for
Advection-diffusion equation. Journal of Phy. Conference Series. 2017;909.
Qiqi Tran, Jinjie Lin. Modified Iterated Crank-Nicolson method with improved Accuracy.
arXiv: 1608.01344 V1 [math.NA].
Simeon Kiprono Mariton. Modified Crank Nicholson Based Methods on the Solution of one
dimensional Heat Equation. Nonlinear Analysis and Differential Equations. 2019;7(1):33-37.
Omowo et al.; ARJOM, 18(10): 1-10, 2022; Article no.ARJOM.81687
Cooper J. Introduction to Partial differential Equation with Matlab. Boston; 1958.
Mitchell AR, Gridffiths DF. A Finite difference method in partial differential equations. John
Wiley and Sons; 1980.
Williams F. Ames, Numerical methods for Partial differential Equations, Academic Press, Inc,
Third Edition; 1992.
Smith GD. Numerical solution of partial differential equations: Finite difference methods.
Clarendon Press, Third Edition, Oxford; 1985.
Grewal BS. Higher Engineering Mathematics. Khanna Publisher, Forty-second Edition; 2012.
John Strikwerda. Finite difference schemes and Partial differential equations. SIAM, Society
for Industrial and Applied Mathematics; 2004.
Iyengar SRK, Jain RK. Numerical methods. New Age International Publisher; 2009.
parabolic equation. International Journal of Scientific Research. 2019;15(6)series 3:60-66.
Crank J, Philis N. A practical method for Numerical Evaluation of solution of partial
differential equation of heat conduction type. Proc. camb. Phil. Soc. 1996;1:50-57.
Omowo BJ, Abhulimen CE. On the stability of Modified Crank-Nicolson method for
Parabolic Partial differential equations. International Journal of Mathematical Sciences and
Optimization: Theory and Application. 2021;6(2):862-873.
Omowo Babajide Johnson, Longe Idowu Oluwaseun. Crank-Nicolson and Modified Crank-
Nicolson scheme for one dimensional parabolic partial differential equation. International
journal of Applied Mathematics and Theoretical Physics. 2020;6(3):35-40.
Omowo BJ, et al. On the Convergence and Stability of Finite Difference Method for Parabolic
Partial Differential Equations. Journal of Advances in Mathematics and Computer Science.
;36 (10):58-67.
Febi Sanjaya, Sudi Mungkasi. A simple but accurate explicit finite difference method for
Advection-diffusion equation. Journal of Phy. Conference Series. 2017;909.
Qiqi Tran, Jinjie Lin. Modified Iterated Crank-Nicolson method with improved Accuracy.
arXiv: 1608.01344 V1 [math.NA].
Simeon Kiprono Mariton. Modified Crank Nicholson Based Methods on the Solution of one
dimensional Heat Equation. Nonlinear Analysis and Differential Equations. 2019;7(1):33-37.
Omowo et al.; ARJOM, 18(10): 1-10, 2022; Article no.ARJOM.81687
Cooper J. Introduction to Partial differential Equation with Matlab. Boston; 1958.
Mitchell AR, Gridffiths DF. A Finite difference method in partial differential equations. John
Wiley and Sons; 1980.
Williams F. Ames, Numerical methods for Partial differential Equations, Academic Press, Inc,
Third Edition; 1992.
Smith GD. Numerical solution of partial differential equations: Finite difference methods.
Clarendon Press, Third Edition, Oxford; 1985.
Grewal BS. Higher Engineering Mathematics. Khanna Publisher, Forty-second Edition; 2012.
John Strikwerda. Finite difference schemes and Partial differential equations. SIAM, Society
for Industrial and Applied Mathematics; 2004.
Iyengar SRK, Jain RK. Numerical methods. New Age International Publisher; 2009.
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