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Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system

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  • Owolabi, Kolade M.
  • Karaagac, Berat

Abstract

This paper focuses on the design and analysis of an efficient numerical method based on the novel implicit finite difference scheme for the solution of the dynamics of reaction-diffusion models. The work replaces the integer first order derivative in time with the Caputo fractional derivative operator. The dynamics of activator-inhibitor as encountered in chemistry, physics and engineering processes, and predator-prey models are two cases addresses in this study. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction-diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve cross-diffusion problem in two-dimensions. In the experimental results, a number of spatiotemporal and chaotic patterns that are related to Turing pattern are observed. It was discovered in the simulation experiments that the species predator-prey model distribute in almost same fashion, while that of the activator-inhibitor dynamics behaved differently regardless of the value of fractional order chosen.

Suggested Citation

  • Owolabi, Kolade M. & Karaagac, Berat, 2020. "Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    • Handle: RePEc:eee:chsofr:v:141:y:2020:i:c:s0960077920306986
    DOI: 10.1016/j.chaos.2020.110302
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    References listed on IDEAS

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    1. Yadav, Swati & Pandey, Rajesh K. & Shukla, Anil K., 2019. "Numerical approximations of Atangana–Baleanu Caputo derivative and its application," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 58-64.
    2. Xue, Lin, 2012. "Pattern formation in a predator–prey model with spatial effect," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(23), pages 5987-5996.
    3. Yadav, Swati & Pandey, Rajesh K., 2020. "Numerical approximation of fractional burgers equation with Atangana–Baleanu derivative in Caputo sense," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    4. Owolabi, Kolade M., 2016. "Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 89-98.
    5. dos S. Silva, F.A. & Viana, R.L. & Lopes, S.R., 2015. "Pattern formation and Turing instability in an activator–inhibitor system with power-law coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 487-497.
    6. Singh, Harendra & Pandey, Rajesh K. & Singh, Jagdev & Tripathi, M.P., 2019. "A reliable numerical algorithm for fractional advection–dispersion equation arising in contaminant transport through porous media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 527(C).
    7. Karaagac, Berat, 2019. "A study on fractional Klein Gordon equation with non-local and non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 218-229.
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    Cited by:

    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. Owolabi, Kolade M. & Pindza, Edson & Atangana, Abdon, 2021. "Analysis and pattern formation scenarios in the superdiffusive system of predation described with Caputo operator," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    3. Shi, Dongyang & Zhang, Sihui, 2024. "Unconditional superconvergence analysis of an energy-stable L1 scheme for coupled nonlinear time-fractional prey-predator equations with nonconforming finite element," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    4. Belmahi, Naziha & Shawagfeh, Nabil, 2021. "A new mathematical model for the glycolysis phenomenon involving Caputo fractional derivative: Well posedness, stability and bifurcation," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).

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