Skip to main content
SpringerLink
Log in

A novel conformable fractional approach to the Brusselator system with numerical simulation

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this study, we delve into a comprehensive analysis of the Brusselator system, combining both analytical and numerical approaches. In summary, our initial approach involves revisiting the classic Brusselator system using a conformable fractional derivative-based approach. Starting from this innovative reformulation, we obtain a nonlinear Volterra-type equation. This transformation allows us to simultaneously demonstrate the existence and uniqueness of the solution, while providing us with the necessary tools to develop an efficient numerical approximation method to solve the problem. Subsequently, we present a numerical simulation based on the Nyström method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

  1. Belousov, B.P.: A Reaction with a Periodic Variation of the Colour. Collection of Abstracts of Scientific Papers, pp. 145–147. Institute of Biological Physics, Academy of Sciences of the USSR, Moscow (1959)

    Google Scholar 

  2. Manohara, G., Kumbinarasaiah, S.: Fibonacci wavelet collocation method for the numerical approximation of fractional order Brusselator chemical model. J. Math. Chem. (2023). https://doi.org/10.1007/s10910-023-01521-4

    Article  Google Scholar 

  3. Sarwar, S., Iqbal, S.: Stability analysis, dynamical behavior and analytical solutions of nonlinear fractional differential system arising in chemical reaction. Chin. J. Phys. 56, 374–384 (2018). https://doi.org/10.1016/j.cjph.2017.11.009

    Article  MathSciNet  Google Scholar 

  4. Kundepudi, D., Prigogine, I.: Modern Thermodynamics: From Heat Engines to Dissipative Structures. Wiley, New York (1998)

    Google Scholar 

  5. Fang, Y., Wang, H.: A Brusselator-based model for the growth of a single cell under nutrient limitation. Sci. Rep. 9(1), 1–10 (2019)

    Google Scholar 

  6. Tsuda, T., Mori, H.: A coupled Brusselator model for the formation of clouds and rain. J. Atmos. Sci. 51(16), 2737–2749 (1994)

    Google Scholar 

  7. Liu, Y., Li, W.: A Brusselator model for the spatial spread of epidemics. J. Theor. Biol. 298, 106–112 (2012)

    Google Scholar 

  8. Nonlinear stability analysis of the full Brusselator reaction–diffusion model. https://link.springer.com/article/10.1134/S0040579514060025

  9. Manaa, S.A., Saeed, R.K., Easif, F.H.: Numerical stability of Brusselator system. Raf. J. Comput. Math. 8, 2 (2011)

    Google Scholar 

  10. Goryunov, V.E.: The Andronov–Hopf bifurcation in a biophysical model of the Belousov reaction. Aut. Control Comput. Sci. 52, 694–699 (2018). https://doi.org/10.3103/S0146411618070118

    Article  Google Scholar 

  11. Sukhtayev, A., Zumbrun, K., Jung, S., et al.: Diffusive stability of spatially periodic solutions of the Brusselator model. Commun. Math. Phys. 358, 1–43 (2018). https://doi.org/10.1007/s00220-017-3056-x

    Article  MathSciNet  Google Scholar 

  12. Prigogine, I., Lefever, R.: Symmetry breaking in irreversible processes. J. Chem. Phys. 54(12), 4648–4654 (1971)

    Google Scholar 

  13. Mathematical Modeling of the Brusselator. https://www3.nd.edu/~powers/mcdowell.pdf

  14. Kaplan, D., Nijhout, H.F.: Oscillatory behavior in a model of the Belousov–Zhabotinsky reaction. J. Chem. Phys. 61(12), 4994–5008 (1976)

    Google Scholar 

  15. Gurevich, Y.M., Melnikov, A.V.: Oscillations in a model of the Belousov–Zhabotinsky reaction. Physica D 1(1), 1–19 (1978)

    Google Scholar 

  16. Cross, M.C., Rasmussen, P.G.: Pattern formation in the Belousov–Zhabotinsky reaction. J. Chem. Phys. 69(7), 3239–3250 (1978)

    Google Scholar 

  17. Grassberger, P., Mandelbrot, B.B.: The strange attractor of the Belousov–Zhabotinsky reaction. Phys. Lett. A 99(1–2), 216–223 (1984)

    Google Scholar 

  18. Singh, P.: Applications of the Brusselator model to chemical, biological, and ecological systems. Prog. Theor. Phys. 70(6), 1779–1792 (1983)

    Google Scholar 

  19. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Frat. Differ. Appl. 1(2), 1–3 (2015). https://doi.org/10.12785/pfda/010201

    Article  Google Scholar 

  20. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)

    Google Scholar 

  21. Atangana, A., Baleanu, D.: Fractional Variational Calculus with Applications in Mechanics. Springer (2017)

    Google Scholar 

  22. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  Google Scholar 

  23. Guebbai, H., Ghiat, M.: New conformable fractional derivative definition for positive and increasing functions and its generalization. Adv. Dyn. Syst. Appl. 11(2), 105–111 (2016)

    MathSciNet  Google Scholar 

  24. Moumen Bekkouche, M., Guebbai, H., Kurulay, M., et al.: A new fractional integral associated with the Caputo–Fabrizio fractional derivative. Rend. Circ. Mat. Palermo II Ser. 70, 1277–1288 (2021). https://doi.org/10.1007/s12215-020-00557-8

    Article  MathSciNet  Google Scholar 

  25. Linz, P.: Analytical and Numerical Methods for Volterra Equations. Society for Industrial Mathematics (1987)

    Google Scholar 

  26. Segni, S., Ghiat, M.: Hamza Guebbai new approximation method for volterra nonlinear integro-differential equation. https://doi.org/10.1142/S1793557119500165

  27. Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  28. Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Approach. Springer, New York (2009)

    Google Scholar 

  29. Gautschi, W.: Numerical Analysis. Springer, New York (2012)

    Book  Google Scholar 

  30. Tair, B., Ghait, M., Guebbai, H., Mohemd, A.Z.: Numerical solution of non-linear volterra integral equation of the first kind. Bol. Soc. Paran. Mat. 41, 1–11 (2023)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées et de Modélisation, Faculté de Mathématiques et de l’Informatique et des Sciences de la Matière, Université 8 Mai 1945 Guelma, B.P. 401, 24000, Guelma, Algeria

    Mohamed Lamine Merikhi, Hamza Guebbai, Noureddine Benrabia & Mohamed Moumen Bekkouche

  2. Faculté des Sciences et de la Technologie, Université Mohamed-Chérif Messaadia Souk Ahras, 41000, Souk Ahras, Algeria

    Noureddine Benrabia

  3. Faculté des Sciences, Université El Oued, 39000, El Oued, Algeria

    Mohamed Moumen Bekkouche

Authors
  1. Mohamed Lamine Merikhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hamza Guebbai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Noureddine Benrabia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mohamed Moumen Bekkouche
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed Lamine Merikhi.

Ethics declarations

Conflict of interest:

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Merikhi, M.L., Guebbai, H., Benrabia, N. et al. A novel conformable fractional approach to the Brusselator system with numerical simulation. J. Appl. Math. Comput. 70, 1707–1721 (2024). https://doi.org/10.1007/s12190-024-02022-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-024-02022-6

Keywords

Mathematics Subject Classification

Search

Navigation