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Pattern formation in a fractional reaction–diffusion system

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  • Gafiychuk, V.V.
  • Datsko, B.Yo.

Abstract

We investigate pattern formation in a fractional reaction–diffusion system. By the method of computer simulation of the model of excitable media with cubic nonlinearity we are able to show structure formation in the system with time and space fractional derivatives. We further compare the patterns obtained by computer simulation with those obtained by simulation of the similar system without fractional derivatives. As a result, we are able to show that nonlinearity plays the main role in structure formation and fractional derivative terms change the transient dynamics. So, when the order of time derivative increases and approaches the value of 1.5, the special structure formation switches to homogeneous oscillations. In the case of space fractional derivatives, the decrease of the order of these derivatives leads to more contrast dissipative structures. The variational principle is used to find the approximate solution of such fractional reaction–diffusion model. In addition, we provide a detailed analysis of the characteristic dissipative structures in the system under consideration.

Suggested Citation

  • Gafiychuk, V.V. & Datsko, B.Yo., 2006. "Pattern formation in a fractional reaction–diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(2), pages 300-306.
    • Handle: RePEc:eee:phsmap:v:365:y:2006:i:2:p:300-306
    DOI: 10.1016/j.physa.2005.09.046
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    References listed on IDEAS

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    4. Buceta, J. & Lindenberg, Katja, 2003. "Patterns in reaction–diffusion systems generated by global alternation of dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 325(1), pages 230-242.
    5. Hilfer, R., 2003. "On fractional diffusion and continuous time random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 329(1), pages 35-40.
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    Citations

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    Cited by:

    1. Lenzi, E.K. & Menechini Neto, R. & Tateishi, A.A. & Lenzi, M.K. & Ribeiro, H.V., 2016. "Fractional diffusion equations coupled by reaction terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 9-16.
    2. Hernandez-Martinez, Eliseo & Valdés-Parada, Francisco & Alvarez-Ramirez, Jose & Puebla, Hector & Morales-Zarate, Epifanio, 2016. "A Green’s function approach for the numerical solution of a class of fractional reaction–diffusion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 133-145.
    3. Gafiychuk, V. & Datsko, B. & Meleshko, V., 2008. "Analysis of fractional order Bonhoeffer–van der Pol oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(2), pages 418-424.
    4. Chen, Juan & Zhou, Hua-Cheng & Zhuang, Bo & Xu, Ming-Hua, 2023. "Active disturbance rejection control to stabilization of coupled delayed time fractional-order reaction–advection–diffusion systems with boundary disturbances and spatially varying coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    5. Macías-Díaz, J.E., 2018. "A numerically efficient Hamiltonian method for fractional wave equations," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 231-248.
    6. A. S. Hendy & R. H. De Staelen, 2020. "Theoretical Analysis (Convergence and Stability) of a Difference Approximation for Multiterm Time Fractional Convection Diffusion-Wave Equations with Delay," Mathematics, MDPI, vol. 8(10), pages 1-20, October.
    7. Povstenko, Y.Z., 2010. "Evolution of the initial box-signal for time-fractional diffusion–wave equation in a case of different spatial dimensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4696-4707.
    8. Gafiychuk, V. & Datsko, B. & Meleshko, V. & Blackmore, D., 2009. "Analysis of the solutions of coupled nonlinear fractional reaction–diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1095-1104.

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