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Analytical Study of Fractional Reaction-Diffusion Brusselator System

Year 2022, Volume: 19 Issue: 2, 113 - 123, 01.11.2022

Abstract

In this paper, a time fractional reaction-diffusion Brusselator system with the Caputo type fractional derivatives, is solved by Laplace Adomian decomposition method (LADM). Two numerical examples supported by graphics, tables, and discussion are provided, by comparing the numerical results obtained with the exact solution when α = β = 1, it is observed that they are in perfect agreement, this confirms the accuracy and smoothness of the current method.

Thanks

The authors are, sincerely, very grateful to the editor and the entire staff of the journal, best wishes for your noble mission.

References

  • [1] K. S. Miller and B. Ross, ”An Introduction to the Fractional Calculus and Fractional Differential Equations,” Wiley, New York, 1993.
  • [2] I. Podlubny, ”Fractional Differential Equations,” Academic Press, New York, 1999.
  • [3] A. Kilbas, H.M. Srivastava and J.J. Trujillo, ”Theory and Application of Fractional Differential Equations,” Amsterdam: Elsevier, 2006.
  • [4] A. Carpinteri and F. Mainardi, ”Fractals and Fractional Calculus in Continuum Mechanics,” Springer Verlag, Wien, New York, 1997.
  • [5] A.U. Rehman, M.B. Riaz, A. Akg¨ul, S.T. Saeed and D .Baleanu, ”Heat and mass transport impact on MHD second grade fluid: A comparative analysis of fractional operators,” Heat Transfer, vol. 50, no. 7, pp. 7042- 7064, 2021.
  • [6] R. Lefever and G. Nicolis, ”Chemical instabilities and sustained oscillations,” J. Theor. Biol., vol. 30, no. 2, pp. 267–284, 1971.
  • [7] J. Tyson, ”Some further studies of nonlinear oscillations in chemical systems,” J. Chem. Phys., vol. 58, pp. 3919–3930, 1973.
  • [8] E. H. Twizell and A. B. Gumel,”A second-order scheme for the Brus-selator reaction–difusion system,” Q. Cao. J. Math. Chem., vol. 26, pp. 297–316, 1999.
  • [9] G. Adomian, ”The diffusion Brusselator equation,” Comput. Math. Appl., vol. 29, no. 2, pp. 1-3, 1995.
  • [10] J. Biazar and Z. Ayati, ”A numerical solution of reaction-diffusion brusselator system by A.D.M,” Journal of Nature Science and Sustainable Technology., vol. 1, no. 2, pp. 263-270, 2007.
  • [11] A. M. Wazwaz, ”The decomposition method applied to systems of partial differential equations and to the reaction–diffusion Brusselator model,” Appl. Math. Comput., vol. 110, no. (2-3), pp. 251–264, 2000.
  • [12] A. W. Teong, ”The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution,” Eng. Anal. Bound. Elem., vol. 27, pp. 897–903, 2003.
  • [13] S. Islam, A. Ali and S. Haq, ”A computational modeling of the behavior of the two-dimensional reactiondiffusion Brusselator system,” App. Math. Model., vol. 34, pp. 3896–3909, 2010.
  • [14] H. Aminikhan and A. Jamalian, ”An Efficient Method for Solving the Brusselator System,” Walailak J Sci and Tech., vol. 10, no. 5, pp. 449-465, 2013.
  • [15] H. Siraj, A. Ihteram and S. N. Kottakkaran, ”A computational study of two-dimensional reaction–diffusion Brusselator system with applications in chemical processes,” Alexandria Engineering Journal, vol. 60, pp. 4381–4392, 2021.
  • [16] J. Singh, M. M. Rashidi, D. Kumar and R. Swroop, ”A fractional model of a dynamical Brusselator reactiondifusion system arising in triple collision and enzymatic reactions,” Nonlinear Engineering., vol. 5, no. 4, pp. 277–285, 2016.
  • [17] R. M. Jena, S. Chakraverty, H. Rezazadeh and D. D. Ganji, ”On the solution of time-fractional dynamical model of Brusselator reaction-diffusion system arising in chemical reactions,” Math Meth Appl Sci., pp. 1–11, 2020.
  • [18] H. Jafari, C. M. Khalique and M. Nazari, ”Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations,” Appl. Math. Lett., vol. 24, no. 11, pp. 1799–1805, 2011.
  • [19] H. Khan, R. Shah, P. Kumam, D. Baleanu and M. Arif, ”Laplace decomposition for solving nonlinear system of fractional order partial differential equations,” Advances in Difference Equations, vol. 2020:537, 2020.
  • [20] O. H. Mohammed and H.A. Salim, ”Computational methods based laplace decomposition for solving nonlinear system of fractional order differential equations,” Alexandria Engineering Journal., vol. 57, no. 4, pp.3549–3557, 2018.
  • [21] R. Shah, H. Khan, M. Arif, P. Kumam, ”Application of Laplace Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations,” Entropy., vol. 21, no. 335, 2019.
  • [22] A. M. Wazwaz, ”The combined Laplace transform–Adomian decomposition method for handling nonlinear Volterra integro–differential equations,” Appl. Math. Comput., vol. 216, no. 4, pp. 1304–1309, 2010.
  • [23] D. Baleanu and H.K. Jassim, ”Exact Solution of Two-dimensional Fractional Partial Differential Equations,” Fractal Fract., vol. 4, no. 21, 2020.
  • [24] V.F. Morales-Delgado, M.A. Taneco-Hern´andez, and J.F. G´omez Aguilar, ”On the solutions of fractional order of evolution equations,” Eur. Phys. J. Plus, vol. 132: 47, 2017.
  • [25] G. Adomian, ”A review of the decomposition method in applied mathematics,” Math. Anal. Appli., vol. 135, no. 2, pp. 501-544, 1988.
  • [26] G. Adomian, ”Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic, Boston. 1994.
  • [27] K. Abbaoui and Y. Cherrault, ”Convergence of Adomians Method Applied to Differential Equations,” Computers and Mathematics with Applications, vol. 28, pp. 103-109, 1994.
  • [28] K. Abbaoui, Y. Cherrault, ”Convergence of Adomians Method Applied Nonlinear Equations,” Mathematical and Computer Modelling, vol. 20, pp. 69-73, 1994.
Year 2022, Volume: 19 Issue: 2, 113 - 123, 01.11.2022

Abstract

References

  • [1] K. S. Miller and B. Ross, ”An Introduction to the Fractional Calculus and Fractional Differential Equations,” Wiley, New York, 1993.
  • [2] I. Podlubny, ”Fractional Differential Equations,” Academic Press, New York, 1999.
  • [3] A. Kilbas, H.M. Srivastava and J.J. Trujillo, ”Theory and Application of Fractional Differential Equations,” Amsterdam: Elsevier, 2006.
  • [4] A. Carpinteri and F. Mainardi, ”Fractals and Fractional Calculus in Continuum Mechanics,” Springer Verlag, Wien, New York, 1997.
  • [5] A.U. Rehman, M.B. Riaz, A. Akg¨ul, S.T. Saeed and D .Baleanu, ”Heat and mass transport impact on MHD second grade fluid: A comparative analysis of fractional operators,” Heat Transfer, vol. 50, no. 7, pp. 7042- 7064, 2021.
  • [6] R. Lefever and G. Nicolis, ”Chemical instabilities and sustained oscillations,” J. Theor. Biol., vol. 30, no. 2, pp. 267–284, 1971.
  • [7] J. Tyson, ”Some further studies of nonlinear oscillations in chemical systems,” J. Chem. Phys., vol. 58, pp. 3919–3930, 1973.
  • [8] E. H. Twizell and A. B. Gumel,”A second-order scheme for the Brus-selator reaction–difusion system,” Q. Cao. J. Math. Chem., vol. 26, pp. 297–316, 1999.
  • [9] G. Adomian, ”The diffusion Brusselator equation,” Comput. Math. Appl., vol. 29, no. 2, pp. 1-3, 1995.
  • [10] J. Biazar and Z. Ayati, ”A numerical solution of reaction-diffusion brusselator system by A.D.M,” Journal of Nature Science and Sustainable Technology., vol. 1, no. 2, pp. 263-270, 2007.
  • [11] A. M. Wazwaz, ”The decomposition method applied to systems of partial differential equations and to the reaction–diffusion Brusselator model,” Appl. Math. Comput., vol. 110, no. (2-3), pp. 251–264, 2000.
  • [12] A. W. Teong, ”The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution,” Eng. Anal. Bound. Elem., vol. 27, pp. 897–903, 2003.
  • [13] S. Islam, A. Ali and S. Haq, ”A computational modeling of the behavior of the two-dimensional reactiondiffusion Brusselator system,” App. Math. Model., vol. 34, pp. 3896–3909, 2010.
  • [14] H. Aminikhan and A. Jamalian, ”An Efficient Method for Solving the Brusselator System,” Walailak J Sci and Tech., vol. 10, no. 5, pp. 449-465, 2013.
  • [15] H. Siraj, A. Ihteram and S. N. Kottakkaran, ”A computational study of two-dimensional reaction–diffusion Brusselator system with applications in chemical processes,” Alexandria Engineering Journal, vol. 60, pp. 4381–4392, 2021.
  • [16] J. Singh, M. M. Rashidi, D. Kumar and R. Swroop, ”A fractional model of a dynamical Brusselator reactiondifusion system arising in triple collision and enzymatic reactions,” Nonlinear Engineering., vol. 5, no. 4, pp. 277–285, 2016.
  • [17] R. M. Jena, S. Chakraverty, H. Rezazadeh and D. D. Ganji, ”On the solution of time-fractional dynamical model of Brusselator reaction-diffusion system arising in chemical reactions,” Math Meth Appl Sci., pp. 1–11, 2020.
  • [18] H. Jafari, C. M. Khalique and M. Nazari, ”Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations,” Appl. Math. Lett., vol. 24, no. 11, pp. 1799–1805, 2011.
  • [19] H. Khan, R. Shah, P. Kumam, D. Baleanu and M. Arif, ”Laplace decomposition for solving nonlinear system of fractional order partial differential equations,” Advances in Difference Equations, vol. 2020:537, 2020.
  • [20] O. H. Mohammed and H.A. Salim, ”Computational methods based laplace decomposition for solving nonlinear system of fractional order differential equations,” Alexandria Engineering Journal., vol. 57, no. 4, pp.3549–3557, 2018.
  • [21] R. Shah, H. Khan, M. Arif, P. Kumam, ”Application of Laplace Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations,” Entropy., vol. 21, no. 335, 2019.
  • [22] A. M. Wazwaz, ”The combined Laplace transform–Adomian decomposition method for handling nonlinear Volterra integro–differential equations,” Appl. Math. Comput., vol. 216, no. 4, pp. 1304–1309, 2010.
  • [23] D. Baleanu and H.K. Jassim, ”Exact Solution of Two-dimensional Fractional Partial Differential Equations,” Fractal Fract., vol. 4, no. 21, 2020.
  • [24] V.F. Morales-Delgado, M.A. Taneco-Hern´andez, and J.F. G´omez Aguilar, ”On the solutions of fractional order of evolution equations,” Eur. Phys. J. Plus, vol. 132: 47, 2017.
  • [25] G. Adomian, ”A review of the decomposition method in applied mathematics,” Math. Anal. Appli., vol. 135, no. 2, pp. 501-544, 1988.
  • [26] G. Adomian, ”Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic, Boston. 1994.
  • [27] K. Abbaoui and Y. Cherrault, ”Convergence of Adomians Method Applied to Differential Equations,” Computers and Mathematics with Applications, vol. 28, pp. 103-109, 1994.
  • [28] K. Abbaoui, Y. Cherrault, ”Convergence of Adomians Method Applied Nonlinear Equations,” Mathematical and Computer Modelling, vol. 20, pp. 69-73, 1994.
There are 28 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Mohamed Zellal 0000-0001-6903-6106

Kacem Belghaba 0000-0003-4372-6330

Publication Date November 1, 2022
Published in Issue Year 2022 Volume: 19 Issue: 2

Cite

APA Zellal, M., & Belghaba, K. (2022). Analytical Study of Fractional Reaction-Diffusion Brusselator System. Cankaya University Journal of Science and Engineering, 19(2), 113-123.
AMA Zellal M, Belghaba K. Analytical Study of Fractional Reaction-Diffusion Brusselator System. CUJSE. November 2022;19(2):113-123.
Chicago Zellal, Mohamed, and Kacem Belghaba. “Analytical Study of Fractional Reaction-Diffusion Brusselator System”. Cankaya University Journal of Science and Engineering 19, no. 2 (November 2022): 113-23.
EndNote Zellal M, Belghaba K (November 1, 2022) Analytical Study of Fractional Reaction-Diffusion Brusselator System. Cankaya University Journal of Science and Engineering 19 2 113–123.
IEEE M. Zellal and K. Belghaba, “Analytical Study of Fractional Reaction-Diffusion Brusselator System”, CUJSE, vol. 19, no. 2, pp. 113–123, 2022.
ISNAD Zellal, Mohamed - Belghaba, Kacem. “Analytical Study of Fractional Reaction-Diffusion Brusselator System”. Cankaya University Journal of Science and Engineering 19/2 (November 2022), 113-123.
JAMA Zellal M, Belghaba K. Analytical Study of Fractional Reaction-Diffusion Brusselator System. CUJSE. 2022;19:113–123.
MLA Zellal, Mohamed and Kacem Belghaba. “Analytical Study of Fractional Reaction-Diffusion Brusselator System”. Cankaya University Journal of Science and Engineering, vol. 19, no. 2, 2022, pp. 113-2.
Vancouver Zellal M, Belghaba K. Analytical Study of Fractional Reaction-Diffusion Brusselator System. CUJSE. 2022;19(2):113-2.