Pattern formation in a fractional reaction–diffusion system

Jun 30th, 2015
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Pattern formation in evolutionary dynamics has been studied as an example of phenomena inherent to many natural systems. It is well known that the reaction–diffusion models are used to study self-organization phenomena in physical, chemical and biological systems (see, for example, Refs. [1–4]). The interest in selforganization phenomena dynamics as a means to explore diversity of the phenomena in nonlinear dissipative systems far from equilibrium has grown exponentially over the past decade. There are several approaches that we use to find solutions for nonlinear dissipative systems. Computer.

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ARTICLE IN PRESSPhysica A 365 (2006) 300306www.elsevier.com/locate/physaPattern formation in a fractional reactiondiffusion systemV.V. Gaychuka,, B.Yo. DatskobaInstitute of Computer Modeling, Cracow University of Technology, ul. Warszawska 24, Cracow 31-155, PolandInstitute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences, Naukova St, 3b, Lviv 79601, UkrainebReceived 28 June 2005; received in revised form 13 September 2005Available online 18 October 2005AbstractWe investigate pattern formation in a fractional reactiondiffusion system. By the method of computer simulation of themodel of excitable media with cubic nonlinearity we are able to show structure formation in the system with time and spacefractional derivatives. We further compare the patterns obtained by computer simulation with those obtained bysimulation of the similar system without fractional derivatives. As a result, we are able to show that nonlinearity plays themain role in structure formation and fractional derivative terms change the transient dynamics. So, when the order of timederivative increases and approaches the value of 1.5, the special structure formation switches to homogeneous oscillations.In the case of space fractional derivatives, the decrease of the order of these derivatives leads to more contrast dissipativestructures. The variational principle is used to nd the approximate solution of such fractional reactiondiffusion

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