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On the Solutions of the Fractional Generalized Gierer–Meinhardt Model

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Fractional Differential Equations (INDAM 2021)

Part of the book series: Springer INdAM Series ((SINDAMS,volume 50))

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Abstract

In this chapter, we consider a time-fractional generalized Gierer–Meinhardt model described by a system of two coupled nonlinear time-fractional reaction–diffusion equations. Solutions are computed by applying a procedure that combines the Lie symmetry analysis with classical numerical methods. Lie symmetries reduce the target system into time-fractional coupled ordinary differential equations. The numerical solutions are determined by introducing the Caputo definition fractional derivative and by using an implicit classical numerical method. Numerical results are presented to validate the effectiveness of the proposed approach and to show, by a comparison with the integer-order case, that the fractional-order model can be considered a reliable generalization of the classical model.

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References

  1. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier (2006)

    MATH  Google Scholar 

  3. Samko, S., Kilbas, A.A., Marichev, O.: Fractional Integrals and Derivatives. Taylor and Francis (1993)

    Google Scholar 

  4. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2004)

    Google Scholar 

  5. Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Physica A, 276, 448–455 (2000)

    Article  MathSciNet  Google Scholar 

  6. Seki, K., Wojcik, M., Tachiya, M.: Fractional reaction-diffusion equation. J. Chem. Phys. 119, 2165 (2003)

    Article  Google Scholar 

  7. Gafiychuk, V., Datsko, B.: Pattern formation in a fractional reaction-diffusion system. Phys. A Stat. Mech. Appl. 365, 300–306 (2006)

    Article  Google Scholar 

  8. Gafiychuk, V.V., Datsko, B.Y.: Spatio-temporal pattern formation in fractional reaction-diffusion systems with indices of different order. Phys. Rev. E 77(6), 066210 (2008)

    Article  MathSciNet  Google Scholar 

  9. Gafiychuk, V., Datsko, B.: Different types of instabilities and complex dynamics in reaction-diffusion systems with fractional derivatives. J. Comput. Nonlin. Dyn. 7, 031001 (2012)

    Article  Google Scholar 

  10. Datsko, B., Gafiychuk, V.: Complex nonlinear dynamics in subdiffusive activator-inhibitor systems. Commun. Nonlinear Sci. Numer. Simul. 17, 1673–1680 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nec, Y.: Explicitly solvable eigenvalue problem and bifurcation delay in sub-diffusive Gierer-Meinhardt model. Eur. J. Appl. Math. 27(5), 699–725 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nec, Y.: Spike solutions in Gierer#x2013; Meinhardt model with a time dependent anomaly exponent. Commun. Nonlinear Sci. Numer. Simul. 54, 267–285 (2018)

    Article  Google Scholar 

  13. Jannelli, A.: Numerical solutions of fractional differential equations arising in engineering sciences. Mathematics 8, 215 (2020)

    Article  Google Scholar 

  14. Jannelli, A.: Adaptive numerical solutions of time-fractional advection-diffusion-reaction equations. Commun. Nonlinear. Sci. Numer. Simul. 105, 106073 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972); Biological Cybernetics 12(1), 30–39 (1972). https://doi.org/10.1007/BF00289234

  16. Henry, B.I., Wearne, S.L.: Existence of turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math. 62(3), 870–887 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Turing pattern formation in fractional activator-inhibitor systems. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 72(2), 026101 (2005)

    Article  MathSciNet  Google Scholar 

  18. Guo, L., Zeng, F., Turner, I., Burrage, K., Karniadakis, G.E.M.: Efficient multistep methods for tempered fractional calculus: Algorithms and simulations. SIAM J. Sci. Comput. 41(4), 2510–2535 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yu, W., Rongpei, Z., Zhen, Zijian, H.: Turing pattern in the fractional Gierer-Meinhardt model. Chin. Phys. B 28(5), 050503 (2019)

    Google Scholar 

  20. Wei, J., Yang, W.: Multi-bump Ground states of the fractional Gierer-Meinhardt system on the real line. J. Dyn. Differ. Equ. 31, 385–417 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Meinhardt, H., Klingler, M.: A model for pattern formation on the shells of molluscs. J. Theor. Biol. 126, 63–89 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Buceta, J., Lindenberg, K.: Switching-induced Turing instability. Phys. Rev. E 66, 046202 (2002)

    Article  MathSciNet  Google Scholar 

  23. Wu, R., Shao, Y., Zhou, Y., Chen, L.: Turing and Hopf bifurcation of Gierer-Meinhardt activator-substrate model. Electron. J. Differ. Equ. 2017(173), 1–19 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Chen, L., Wu, R., Xu, Y.: Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme. Discrete Contin. Dyn. Syst. B (2020). https://doi.org/10.3934/dcdsb.2021132

  25. Jannelli, A., Ruggieri, M., Speciale, M.P.: Numerical solutions of space fractional advection-diffusion equation with source term. AIP Confer. Proc. 2116, 280007 (2019)

    Article  MATH  Google Scholar 

  26. Jannelli, A., Ruggieri, M., Speciale, M.P.: Numerical solutions of space fractional advection–diffusion equation, with nonlinear source term. Appl. Num. Math. 155, 93–102 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jannelli, A., Speciale, M.P.: On the numerical solutions of coupled nonlinear time-fractional reaction–diffusion equations. AIMS Math. 6(8), 9109–9125 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jannelli, A., Speciale, M.P.: Exact and numerical solutions of two-dimensional time–fractional diffusion-reaction equation through the lie symmetries. Nonlinear Dynamics 105, 2375–2385 (2021)

    Article  Google Scholar 

  29. Jannelli, A., Speciale, M.P.: Comparison between Solutions of two-dimensional time-fractional diffusion-reaction equation through the lie symmetries. Atti della Accademia Peloritana dei Pericolanti 99, A4 (2021)

    MathSciNet  Google Scholar 

  30. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Continuous transformation groups of fractional differential equations, Vestn. USATU 9, 125–35 (2007)

    Google Scholar 

  31. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Group-invariant solutions of fractional differential equations. Nonlinear Sci. Complex. 51–59 (2011)

    Google Scholar 

  32. Vu, K.T., Jefferson, G.F., Carminati, J.: Finding generalized symmetries of differential equations using the MAPLE package DESOLVII. Comput. Phys. Commun. 183, 1044–1054 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Jefferson, G.F., Carminati, J.: ASP: Automated symbolic computation of approximate symmetries of differential equations. Comput. Phys. Commun. 184, 1045–1063 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Garrappa, R.: Trapezoidal methods for fractional differential equations: Theoretical and computational aspects. Math. Comput. Simul. 110, 96–112 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

A. J. is a member of GNCS-INdAM Research Group. M.P. S. is a member of GNFM-INdAM Research Group.

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Authors and Affiliations

  1. Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Messina, Italy

    Alessandra Jannelli & Maria Paola Speciale

Authors
  1. Alessandra Jannelli
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  2. Maria Paola Speciale
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Corresponding authors

Correspondence to Alessandra Jannelli or Maria Paola Speciale .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Salerno, Fisciano, Salerno, Italy

    Angelamaria Cardone

  2. Department of Science and High Technology, University of Insubria, Como, Italy

    Marco Donatelli

  3. Department of Mathematics, University of Pisa, Pisa, Italy

    Fabio Durastante

  4. Department of Mathematics, University of Bari Aldo Moro, Bari, Italy

    Roberto Garrappa

  5. Department of Science and High Technology, University of Insubria, Como, Italy

    Mariarosa Mazza

  6. Department of Electrical and Information Engineering, Polytechnic University of Bari, Bari, Italy

    Marina Popolizio

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Jannelli, A., Speciale, M.P. (2023). On the Solutions of the Fractional Generalized Gierer–Meinhardt Model. In: Cardone, A., Donatelli, M., Durastante, F., Garrappa, R., Mazza, M., Popolizio, M. (eds) Fractional Differential Equations. INDAM 2021. Springer INdAM Series, vol 50. Springer, Singapore. https://doi.org/10.1007/978-981-19-7716-9_6

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