Numerical patterns in system of integer and non-integer order derivatives

Kolade M. Owolabi

Chaos, Solitons & Fractals, 2018, vol. 115, issue C, 143-153

Abstract: This research work contributes to the formation of spatial patterns in fractional-order reaction-diffusion systems. The classical second-order partial derivatives in such systems are replaced with the Riemann–Liouville fractional derivative of order α ∈ (1, 2]. We equally propose a novel numerical scheme for the approximation in space, and the resulting system of equations is advance in time with the improved fourth-order exponential time differencing method. Mathematical analysis of general two-component integer and non-integer order derivatives are provided. To guarantee the correct choice of the parameters in the main dynamics, we carry-out their linear stability analysis. Theorems regarding the local-stability and the conditions for a Hopf-bifurcation to occur are also provided. The proposed numerical method is applied to solve two non-integer-order models, namely the biological (predator-prey) and chemical (activator-inhibitor) systems. We observed some amazing patterns that are completely missing in the classical case at different values of fractional power α in high dimensions that evolve in fractional reaction-diffusion equations.

Keywords: Exponential time-integrator; Fractional reaction-diffusion; Nonlinear PDEs; Numerical simulations; Hopf-bifurcation (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (11)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:115:y:2018:i:c:p:143-153

DOI: 10.1016/j.chaos.2018.08.010

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