Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems

Khaled Abdalla Ishag; Mohamed Ahmed Abdallah; Abulfida Mohamed Ahmed

doi:doi:10.11648/j.acm.20221102.12

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Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems

Khaled Abdalla Ishag, Mohamed Ahmed Abdallah, Abulfida Mohamed Ahmed

Published in Applied and Computational Mathematics (Volume 11, Issue 2)

Received: 22 March 2022 Accepted: 11 April 2022 Published: 20 April 2022

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Abstract

This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the sum of an infinite series. Which converges rapidly to exact solution. The Homotopy Perturbation Method is no need to use Adomian's polynomials to calculate the nonlinear terms; we test the proposed method to solve nonlinear fractional systems of redaction diffusion equations in one dimension, two dimensions and three dimensions. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with ordinary derivative order index α₁=α₂=1 for nonlinear fractional reaction diffusion systems. Approximate solutions for different values of fractional derivatives index α₁=0.5 and α₂=0.5 together with non-fractional derivative index α₁=1 and α₂=1 and absolute errors are represented graphically in two and three dimensions. In addition, the graphical represented the solutions, which had been given by MATLAB program. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods.

Published in	Applied and Computational Mathematics (Volume 11, Issue 2)
DOI	10.11648/j.acm.20221102.12
Page(s)	48-55
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), 2022. Published by Science Publishing Group

Keywords

Fractional Calculus, Diffusion Equations, Homotopy Perturbation Method, Approximate Solutions

References

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[2]	Mohamed Ahmed Abdalla, Asymptotic behavior of solution of a periodic mutualistic system, 2016.
[3]	Christina Kuttler, Reaction – diffusion equations with applications, 2011.
[4]	Lokenath Debnath. ''Nonlinear partial differential equations for scientists and engineers'', Springer Science and Business, 2005.
[5]	Safyan Mukhtar, Salah Abuasad, Ishak Hashim, Samsul Ariffin Abdul Karim, Effective method for solving different types of nonlinear fractional Burger's equation, Mathematics 2020, 8, 729, doi: 10.3390/math8050729.
[6]	Khaled Abdalla Ishag, The techniques for solving fractional Burger's equations, Asian Journal of Pure and Applied Mathematics, Vol. 4 (2022).
[7]	Khalid Suliman Aboodh and Abu baker Ahmed. On the application of homotopy analysis method to fractional equation, Journal of the Faculty of Science and Technology, 7 (2020) 1-18.
[8]	Abuasad, S.; Yildirim, A.; Hashim, I.; Abdul Karim, S. A.; Gómez-Aguilar, J. Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination. Int. J. Environ. Res. Public Health 2019, 16, 973.
[9]	Khaled A. Ishag, Faris Azhari Okasha, Homotopy perturbation transform method for extensile beam equations, FES journal of engineering sciences, Vol. 8, No. 1, 2019.
[10]	Abuasad, S.; Hashim, I. Homotopy decomposition method for solving one-dimensional time-fractional diffusion equation. In AIP Conference Proceedings; AIP Publishing: Melville, NY, USA, 2018, Volume 1940; p. 020126.
[11]	Abuasad, S.; Hashim, I. Homotopy Decomposition Method for Solving Higher-Order Time- Fractional Diffusion Equation via Modified Beta Derivative. Sains Malays. 2018, 47, 2899–2905.
[12]	Abuasad, S.; Hashim, I.; Abdul Karim, S. A. Modified Fractional Reduced Differential Transform Method for the Solution of Multiterm Time-Fractional Diffusion Equations. Adv. Math. Phys. 2019, 5703916.
[13]	J. F. Gomez, homotopy perturbation transform method for nonlinear differential equation involving to fractional operator with exponential kernel, Advances in difference equation, 2017.
[14]	Xin, B.; Peng, W.; Kwon, Y.; Liu, Y. Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk. Adv. Differ. Equ. 2019, 138.
[15]	Srivastava, H.; Gunerhan, H. Analytical and approximate solutions of fractional-order susceptible-infected-recovered epidemic model of childhood disease. Math. Methods Appl. Sci. 2019, 42, 935–941.
[16]	El-Sayed, A. M. A.; Nour, H. M.; Elsaid, A.; Matouk, A. E.; Elsonbaty, A. Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional-order hyperchaotic system. Appl. Math. Model. 2016, 40, 3516–3534.

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APA Style

Khaled Abdalla Ishag, Mohamed Ahmed Abdallah, Abulfida Mohamed Ahmed. (2022). Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems. Applied and Computational Mathematics, 11(2), 48-55. https://doi.org/10.11648/j.acm.20221102.12

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ACS Style

Khaled Abdalla Ishag; Mohamed Ahmed Abdallah; Abulfida Mohamed Ahmed. Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems. Appl. Comput. Math. 2022, 11(2), 48-55. doi: 10.11648/j.acm.20221102.12

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AMA Style

Khaled Abdalla Ishag, Mohamed Ahmed Abdallah, Abulfida Mohamed Ahmed. Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems. Appl Comput Math. 2022;11(2):48-55. doi: 10.11648/j.acm.20221102.12

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@article{10.11648/j.acm.20221102.12,
  author = {Khaled Abdalla Ishag and Mohamed Ahmed Abdallah and Abulfida Mohamed Ahmed},
  title = {Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {2},
  pages = {48-55},
  doi = {10.11648/j.acm.20221102.12},
  url = {https://doi.org/10.11648/j.acm.20221102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221102.12},
  abstract = {This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the sum of an infinite series. Which converges rapidly to exact solution. The Homotopy Perturbation Method is no need to use Adomian's polynomials to calculate the nonlinear terms; we test the proposed method to solve nonlinear fractional systems of redaction diffusion equations in one dimension, two dimensions and three dimensions. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with ordinary derivative order index α1=α2=1 for nonlinear fractional reaction diffusion systems. Approximate solutions for different values of fractional derivatives index α1=0.5 and α2=0.5 together with non-fractional derivative index α1=1 and α2=1 and absolute errors are represented graphically in two and three dimensions. In addition, the graphical represented the solutions, which had been given by MATLAB program. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods.},
 year = {2022}
}

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TY  - JOUR
T1  - Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems
AU  - Khaled Abdalla Ishag
AU  - Mohamed Ahmed Abdallah
AU  - Abulfida Mohamed Ahmed
Y1  - 2022/04/20
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221102.12
DO  - 10.11648/j.acm.20221102.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 48
EP  - 55
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221102.12
AB  - This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the sum of an infinite series. Which converges rapidly to exact solution. The Homotopy Perturbation Method is no need to use Adomian's polynomials to calculate the nonlinear terms; we test the proposed method to solve nonlinear fractional systems of redaction diffusion equations in one dimension, two dimensions and three dimensions. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with ordinary derivative order index α1=α2=1 for nonlinear fractional reaction diffusion systems. Approximate solutions for different values of fractional derivatives index α1=0.5 and α2=0.5 together with non-fractional derivative index α1=1 and α2=1 and absolute errors are represented graphically in two and three dimensions. In addition, the graphical represented the solutions, which had been given by MATLAB program. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods.
VL  - 11
IS  - 2
ER  -

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Author Information

Khaled Abdalla Ishag

Department of Basic Science, Faculty of Engineering Science, Omdurman Islamic University, Omdurman, Sudan
Mohamed Ahmed Abdallah

Department of Basic Science, Faculty of Engineering, University of Sinnar, Sinnar, Sudan
Abulfida Mohamed Ahmed

Department of Mathematics, Faculty of Education, University of Nyala, Nyala, Sudan

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APA Style

Khaled Abdalla Ishag, Mohamed Ahmed Abdallah, Abulfida Mohamed Ahmed. (2022). Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems. Applied and Computational Mathematics, 11(2), 48-55. https://doi.org/10.11648/j.acm.20221102.12

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ACS Style

Khaled Abdalla Ishag; Mohamed Ahmed Abdallah; Abulfida Mohamed Ahmed. Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems. Appl. Comput. Math. 2022, 11(2), 48-55. doi: 10.11648/j.acm.20221102.12

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AMA Style

Khaled Abdalla Ishag, Mohamed Ahmed Abdallah, Abulfida Mohamed Ahmed. Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems. Appl Comput Math. 2022;11(2):48-55. doi: 10.11648/j.acm.20221102.12

Copy | Download

@article{10.11648/j.acm.20221102.12,
  author = {Khaled Abdalla Ishag and Mohamed Ahmed Abdallah and Abulfida Mohamed Ahmed},
  title = {Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {2},
  pages = {48-55},
  doi = {10.11648/j.acm.20221102.12},
  url = {https://doi.org/10.11648/j.acm.20221102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221102.12},
  abstract = {This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the sum of an infinite series. Which converges rapidly to exact solution. The Homotopy Perturbation Method is no need to use Adomian's polynomials to calculate the nonlinear terms; we test the proposed method to solve nonlinear fractional systems of redaction diffusion equations in one dimension, two dimensions and three dimensions. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with ordinary derivative order index α1=α2=1 for nonlinear fractional reaction diffusion systems. Approximate solutions for different values of fractional derivatives index α1=0.5 and α2=0.5 together with non-fractional derivative index α1=1 and α2=1 and absolute errors are represented graphically in two and three dimensions. In addition, the graphical represented the solutions, which had been given by MATLAB program. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems
AU  - Khaled Abdalla Ishag
AU  - Mohamed Ahmed Abdallah
AU  - Abulfida Mohamed Ahmed
Y1  - 2022/04/20
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221102.12
DO  - 10.11648/j.acm.20221102.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 48
EP  - 55
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221102.12
AB  - This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the sum of an infinite series. Which converges rapidly to exact solution. The Homotopy Perturbation Method is no need to use Adomian's polynomials to calculate the nonlinear terms; we test the proposed method to solve nonlinear fractional systems of redaction diffusion equations in one dimension, two dimensions and three dimensions. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with ordinary derivative order index α1=α2=1 for nonlinear fractional reaction diffusion systems. Approximate solutions for different values of fractional derivatives index α1=0.5 and α2=0.5 together with non-fractional derivative index α1=1 and α2=1 and absolute errors are represented graphically in two and three dimensions. In addition, the graphical represented the solutions, which had been given by MATLAB program. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods.
VL  - 11
IS  - 2
ER  -

Copy | Download