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Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems

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  • Owolabi, Kolade M.

Abstract

The aim of this paper is to examine pattern formation in the sub— and super-diffusive scenarios and compare it with that of classical or standard diffusive processes in two-component fractional reaction-diffusion systems that modeled a predator-prey dynamics. The focus of the work concentrates on the use of two separate mathematical techniques, we formulate a Fourier spectral discretization method as an efficient alternative technique to solve fractional reaction-diffusion problems in higher-dimensional space, and later advance the resulting systems of ODEs in time with the adaptive exponential time-differencing solver. Obviously, the fractional Fourier approach is able to achieve spectral convergence up to machine precision regardless of the fractional order α, owing to the fact that our approach is able to give full diagonal representation of the fractional operator. The complexity of the dynamics in this system is theoretically discussed and graphically displayed with some examples and numerical simulations in one, two and three dimensions.

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  • 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.
    • Handle: RePEc:eee:chsofr:v:93:y:2016:i:c:p:89-98
    DOI: 10.1016/j.chaos.2016.10.005
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    1. Meerschaert, Mark M. & Mortensen, Jeff & Wheatcraft, Stephen W., 2006. "Fractional vector calculus for fractional advection–dispersion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 181-190.
    2. Kolade M. Owolabi & Kailash C. Patidar, 2015. "Existence and Permanence in a Diffusive KiSS Model with Robust Numerical Simulations," International Journal of Differential Equations, Hindawi, vol. 2015, pages 1-8, December.
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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. & Atangana, Abdon, 2018. "Chaotic behaviour in system of noninteger-order ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 362-370.
    3. Zahra, W.K. & Nasr, M.A. & Van Daele, M., 2019. "Exponentially fitted methods for solving time fractional nonlinear reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 468-490.
    4. Owolabi, Kolade M. & Karaagac, Berat, 2020. "Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    5. Owolabi, Kolade M. & Atangana, Abdon, 2017. "Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 111-119.
    6. Gómez-Aguilar, J.F., 2018. "Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 52-75.
    7. Alqhtani, Manal & Owolabi, Kolade M. & Saad, Khaled M., 2022. "Spatiotemporal (target) patterns in sub-diffusive predator-prey system with the Caputo operator," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    8. Ávalos-Ruiz, L.F. & Zúñiga-Aguilar, C.J. & Gómez-Aguilar, J.F. & Escobar-Jiménez, R.F. & Romero-Ugalde, H.M., 2018. "FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 177-189.
    9. Owolabi, Kolade M. & Atangana, Abdon, 2019. "Computational study of multi-species fractional reaction-diffusion system with ABC operator," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 280-289.
    10. Solís-Pérez, J.E. & Gómez-Aguilar, J.F. & Atangana, A., 2019. "A fractional mathematical model of breast cancer competition model," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 38-54.
    11. Owolabi, Kolade M. & Atangana, Abdon, 2019. "Mathematical analysis and computational experiments for an epidemic system with nonlocal and nonsingular derivative," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 41-49.
    12. Pakhare, Sumit S. & Daftardar-Gejji, Varsha & Badwaik, Dilip S. & Deshpande, Amey & Gade, Prashant M., 2020. "Emergence of order in dynamical phases in coupled fractional gauss map," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    13. Khan, Hasib & Gómez-Aguilar, J.F. & Khan, Aziz & Khan, Tahir Saeed, 2019. "Stability analysis for fractional order advection–reaction diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 737-751.
    14. Owolabi, Kolade M., 2018. "Numerical patterns in reaction–diffusion system with the Caputo and Atangana–Baleanu fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 160-169.
    15. Owolabi, Kolade M., 2020. "High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    16. Yu, Hao & Wu, Boying & Zhang, Dazhi, 2018. "A generalized Laguerre spectral Petrov–Galerkin method for the time-fractional subdiffusion equation on the semi-infinite domain," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 96-111.
    17. Alqhtani, Manal & Owolabi, Kolade M. & Saad, Khaled M. & Pindza, Edson, 2022. "Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    18. Rayal, Ashish & Ram Verma, Sag, 2020. "Numerical analysis of pantograph differential equation of the stretched type associated with fractal-fractional derivatives via fractional order Legendre wavelets," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).

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