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NJIT Applied Mathematics Colloquium

Friday, April 8 2011, 11:30am
Cullimore Lecture Hall II
New Jersey Institute of Technology

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Front propagation in anomalous diffusion-reaction systems

Alexander Nepomnyaschy

Technion

Reaction-diffusion systems describe numerous phenomena in nature. It has been recently understood that many diffusion processes are described by models of anomalous diffusion, which explain the observation of anomalously fast (superdiffusion) or slow (subdiffusion) growth of displacement moments of corresponding random walks. These models include spatial non-locality or/and temporal memory and involve integral operators (e.g., fractional derivatives) in addition to differential operators. The interplay between the anomalous diffusion and the reactions is not yet well understood. In the present talk, we analyze the propagation of reaction fronts in systems with anomalous diffusion. In the case of reaction-superdiffusion equations (spatially non-local equations), we start with the exactly solvable case where the reaction term is a discontinuous piecewise linear function. Applying the Fourier transform, we find traveling fronts and pulses, and discuss the effect of superdiffusion on the solutions. Specific problems that we consider include FitzHugh-Nagumo equations, domain wall pinning, and systems of waves. Also, we investigate the dynamics of fronts governed by a superdiffusive Allen-Cahn equation in one and two dimensions. In the case of reaction-subdiffusion equations (systems with memory), it is necessary to distinguish between the cases of subdiffusion limited and activation limited reactions. In the former case, the system is governed by models with a fractional time derivative, and the front is described by a traveling wave solution. In the latter case, the system is subject to "aging", which is described by integro-differential equations with two time variables, and the velocity of the front decreases with time.