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Research Papers

Numerical Simulation of Noninteger Order System in Subdiffusive, Diffusive, and Superdiffusive Scenarios

[+] Author and Article Information
Kolade M. Owolabi

Institute for Groundwater Studies,
Faculty of Natural and Agricultural Sciences,
University of the Free State,
Bloemfontein 9300, South Africa
e-mails: mkowolax@yahoo.com;
kmowolabi@futa.edu.ng

Abdon Atangana

Institute for Groundwater Studies,
Faculty of Natural and Agricultural Sciences,
University of the Free State,
Bloemfontein 9300, South Africa

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 3, 2016; final manuscript received October 28, 2016; published online December 5, 2016. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 12(3), 031010 (Dec 05, 2016) (7 pages) Paper No: CND-16-1219; doi: 10.1115/1.4035195 History: Received May 03, 2016; Revised October 28, 2016

In this work, we investigate both the mathematical and numerical studies of the fractional reaction–diffusion system consisting of spatial interactions of three components’ species. Our main result is based on the analysis of the model for linear stability. Mathematical analysis of the main equation shows that the dynamical system is both locally and globally asymptotically stable. We further propose a theorem which guarantees the existence and permanence of the three species. We formulate a viable numerical methods in space and time. By adopting the Fourier spectral approach to discretize in space, the issue of stiffness associated with the fractional-order spatial derivatives in such system is removed. The resulting system of ordinary differential equations (ODEs) is advanced with the exponential time-differencing method of ADAMS-type. The complexity of the dynamics in the system which we discussed theoretically are numerically presented through some numerical simulations in 1D, 2D, and 3D to address the points and queries that may naturally arise.

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Figures

Grahic Jump Location
Fig. 1

Surface plot showing the distribution of species for reaction–diffusion system (1). The first and second rows areobtained at t = 10 and t = 20, respectively. The parameter values and initial data: η=1.5, D1=0.7,D2=0.5,D3=0.2, u0(x)=0.53x+0.47 sin(−1.5πx),v0(x)=1+sin(2πx),w0(x)=ŵ+10−8(x−1200)(x−2800), ŵ=1/30. Other parameter values as in Eq.(16).

Grahic Jump Location
Fig. 2

Different chaotic behaviors of the species arising from the perturbation of the cropping rate αi,i=1,2, at different instants of time with fractional order η=0.5. Other parameters and initial condition are as used in Fig. 1. Take note of variation in the amplitudes.

Grahic Jump Location
Fig. 3

Contour plots of Eq. (1) in 2D indicating fractional power effects at different instants of η=0.5,1.5 which correspond to subdiffusive and superdiffusive scenarios for the upper and lower-rows, respectively. Other parameters are as fixed in Eq. (18).

Grahic Jump Location
Fig. 4

Numerical simulations of system (1) with kinetics (2) in 3D showing the effects of fractional power η at different instances of η=0.5, η=2.0, and η=1.5 which correspond to the subdiffusive, diffusive, and superdiffusive cases in rows 1–3, respectively. Other parameters are given in Eq. (18).

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