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Analysis of the solutions of coupled nonlinear fractional reaction–diffusion equations

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  • Gafiychuk, V.
  • Datsko, B.
  • Meleshko, V.
  • Blackmore, D.

Abstract

This paper is concerned with analysis of coupled fractional reaction–diffusion equations. As an example, the reaction–diffusion model with cubic nonlinearity and Brusselator model are considered. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. Computer simulation and analytical methods are used to analyze possible solutions for a linearized system. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. It is shown that an increase of the fractional derivative index leads to periodic solutions which become stochastic as the index approaches the value of 2. It is established by computer simulation that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary condition. The characteristic features of these solutions consist in the transformation of the steady state dissipative structures to homogeneous oscillations or spatio-temporal structures at certain values of the fractional index.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:3:p:1095-1104
    DOI: 10.1016/j.chaos.2008.04.039
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    References listed on IDEAS

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    1. Yu, Rui & Zhang, Hongqing, 2006. "New function of Mittag–Leffler type and its application in the fractional diffusion-wave equation," Chaos, Solitons & Fractals, Elsevier, vol. 30(4), pages 946-955.
    2. 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.
    3. Momani, Shaher & Odibat, Zaid, 2007. "Numerical comparison of methods for solving linear differential equations of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1248-1255.
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    1. Caicedo, Alejandro & Cuevas, Claudio & Mateus, Éder & Viana, Arlúcio, 2021. "Global solutions for a strongly coupled fractional reaction-diffusion system in Marcinkiewicz spaces," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    2. Li, Shanwei & Maimaiti, Yimamu, 2025. "Stability and bifurcation analysis of a time-order fractional model for water-plants: Implications for vegetation pattern formation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 234(C), pages 342-358.
    3. Huina Zhang & Wenjie Gao, 2014. "Existence and Uniqueness Results for a Coupled System of Nonlinear Fractional Differential Equations with Antiperiodic Boundary Conditions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    4. Heydari, M.H. & Atangana, A., 2019. "A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 339-348.
    5. Josefa Caballero & Mohamed Abdalla Darwish & Kishin Sadarangani & Wafa M. Shammakh, 2014. "Existence Results for a Coupled System of Nonlinear Fractional Hybrid Differential Equations with Homogeneous Boundary Conditions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    6. Zeeshan Ali & Poom Kumam & Kamal Shah & Akbar Zada, 2019. "Investigation of Ulam Stability Results of a Coupled System of Nonlinear Implicit Fractional Differential Equations," Mathematics, MDPI, vol. 7(4), pages 1-26, April.

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