561 research outputs found
Fast-slow asymptotic for semi-analytical ignition criteria in FitzHugh-Nagumo system
We study the problem of initiation of excitation waves in the FitzHugh-Nagumo
model. Our approach follows earlier works and is based on the idea of
approximating the boundary between basins of attraction of propagating waves
and of the resting state as the stable manifold of a critical solution. Here,
we obtain analytical expressions for the essential ingredients of the theory by
singular perturbation using two small parameters, the separation of time scales
of the activator and inhibitor, and the threshold in the activator's kinetics.
This results in a closed analytical expression for the strength-duration curve.Comment: 10 pages, 5 figures, as accepted to Chaos on 2017/06/2
Small noise asymptotic expansions for stochastic PDE's driven by dissipative nonlinearity and L\'evy noise
We study a reaction-diffusion evolution equation perturbed by a space-time
L\'evy noise. The associated Kolmogorov operator is the sum of the
infinitesimal generator of a -semigroup of strictly negative type acting
in a Hilbert space and a nonlinear term which has at most polynomial growth, is
non necessarily Lipschitz and is such that the whole system is dissipative.
The corresponding It\^o stochastic equation describes a process on a Hilbert
space with dissi- pative nonlinear, non globally Lipschitz drift and a L\'evy
noise. Under smoothness assumptions on the non-linearity, asymptotics to all
orders in a small parameter in front of the noise are given, with detailed
estimates on the remainders.
Applications to nonlinear SPDEs with a linear term in the drift given by a
Laplacian in a bounded domain are included. As a particular case we provide the
small noise asymptotic expansions for the SPDE equations of FitzHugh Nagumo
type in neurobiology with external impulsive noise.Comment: 29 page
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator
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