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Front propagation in hyperbolic fractional reaction-diffusion equations.

Vicenç Méndez1, Vicente Ortega-Cejas

  • 1Departament de Medicina, Facultat de Ciències de la Salut, Universitat Internacional de Catalunya, c/ Josep Trueta s/n, E-08190 Sant Cugat del Vallès, Barcelona, Spain.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2005
PubMed
Summary

This study analyzes a hyperbolic reaction-diffusion equation with fractional time derivatives. It reveals that under specific conditions, wave fronts exhibit unphysical faster travel in subdiffusive transport.

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Transport Phenomena

Background:

  • Subdiffusive transport processes are often modeled using continuous-time random walks with Lévy waiting time distributions.
  • Reaction-diffusion equations are fundamental in describing phenomena with both reaction and diffusion kinetics.

Purpose of the Study:

  • To investigate the behavior of wave front propagation in a hyperbolic reaction-diffusion equation with time fractional derivatives.
  • To analyze the linear speed selection of these wave fronts under subdiffusive conditions.

Main Methods:

  • Utilizing a continuous-time random walk scheme with a Lévy waiting time distribution.
  • Analyzing a hyperbolic reaction-diffusion equation incorporating time fractional derivatives.
  • Examining the linear speed selection of wave fronts.

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Main Results:

  • Identified conditions where reaction-diffusion dimensionless number and fractional index lead to unusual wave front behavior.
  • Observed wave fronts traveling faster in the subdiffusive regime compared to the diffusive regime.
  • Characterized an unphysical speed phenomenon in fractional reaction-diffusion systems.

Conclusions:

  • The study highlights potential unphysical outcomes in subdiffusive reaction-diffusion models.
  • Findings suggest careful consideration of model parameters and transport regimes is crucial.
  • The research contributes to understanding complex dynamics in fractional calculus applications.