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Evolution of reaction-diffusion patterns in infinite and bounded domains

Author

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  • Hassan, S.A.
  • Kuperman, M.N.
  • Wio, H.S.
  • Zanette, D.H.

Abstract

We introduce a semi-analytical method to study the evolution of spatial structures in reaction-diffusion systems. It consists in writing an integral equation for the relevant densities, from the propagator of the linear part of the evolution operator. In order to test the method, we perform an exhaustive study of a one-dimensional reaction-diffusion model associated to an electrical device - the ballast resistor. We consider the evolution of step and bubble-shaped initial density profiles in free space as well as in a semi-infinite domain with Dirichlet and Neumann boundary conditions. The piecewise-linear form of the reaction term, which preserves the basic ingredients of more complex nonlinear models, makes it possible to obtain exact wave-front solutions in free space and stationary solutions in the bounded domain. Short and long-time behaviour can also be analytically studied, whereas the evolution at intermediate times is analyzed by numerical techniques. We paid particular attention to the features introduced in the evolution by boundary conditions.

Suggested Citation

  • Hassan, S.A. & Kuperman, M.N. & Wio, H.S. & Zanette, D.H., 1994. "Evolution of reaction-diffusion patterns in infinite and bounded domains," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 206(3), pages 380-400.
    • Handle: RePEc:eee:phsmap:v:206:y:1994:i:3:p:380-400
    DOI: 10.1016/0378-4371(94)90313-1
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    Cited by:

    1. Portesi, M & Plastino, A, 1996. "Generalized entropy as a measure of quantum uncertainty," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 225(3), pages 412-430.
    2. Kaniadakis, G. & Quarati, P., 1997. "Polynomial expansion of diffusion and drift coefficients for classical and quantum statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 237(1), pages 229-239.
    3. Pennini, F. & Plastino, A.R. & Plastino, A., 1997. "Dynamical evolution and Tsallis generalized quantum thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 235(3), pages 388-406.
    4. Pennini, F. & Plastino, A. & Plastino, A.R., 1996. "Tsallis nonextensive thermostatistics, Pauli principle and the structure of the Fermi surface," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 234(1), pages 471-479.
    5. Gamero, L.G. & Plastino, A. & Torres, M.E., 1997. "Wavelet analysis and nonlinear dynamics in a nonextensive setting," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 246(3), pages 487-509.
    6. Andricioaei, Ioan & Straub, John E., 1997. "An efficient Monte Carlo algorithm for overcoming broken ergodicity in the simulation of spin systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 247(1), pages 553-558.
    7. Pennini, F. & Plastino, A., 1997. "Fisher's information measure in a Tsallis' nonextensive setting and its application to diffusive process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 247(1), pages 559-569.
    8. Plastino, A.R. & Plastino, A., 1995. "Non-extensive statistical mechanics and generalized Fokker-Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 222(1), pages 347-354.

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