Mathematical modeling of time fractional reaction-diffusion systems

  • Authors:

  • V. Gafiychuk;B. Datsko;V. Meleshko

  • Affiliations:

  • Institute of Computer Modeling, Krakow University of Technology, 24 Warszawska Street, Krakow 31155, Poland and Physics Department, New York City College of Technology, CUNY, 300 Jay Street, NY, N ...;Institute of Applied Problem of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Naukova Street 3 B, Lviv 79053, Ukraine;Institute of Applied Problem of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Naukova Street 3 B, Lviv 79053, Ukraine

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2008

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Abstract

We study a fractional reaction-diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction-diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.