Exact solutions in front propagation problems with superdiffusion

Volpert, Vladimir A.; Nec, Yana; Nepomnyashchy, Alexander

doi:10.1016/j.physd.2009.10.011

Cited by 34 publications

(28 citation statements)

References 36 publications

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“…The travelling wave solution w(z) that describes a front between two stable phases w = u ± with the front velocity c(μ) has been found; a power-law asymptotics, similar to (2.5), has been revealed. [34]). …”

Section: (I) Travelling Wave Solutionsmentioning

confidence: 99%

“…The propagation of fronts in a bistable superdiffusive system governed by the asymmetric model (2.6) with the reaction function (2.3) has been considered in recent studies [34,41].…”

Section: (I) Travelling Wave Solutionsmentioning

confidence: 99%

“…The phenomenon of domain wall pinning was studied in Volpert et al [34] in the framework of the model with a piecewise linear reaction function. A travelling wave stopped by a localized inhomogeneity was described by the equation…”

Section: (Iii) Pinning Of Frontsmentioning

confidence: 99%

“…A superdiffusive version of the FitzHugh-Nagumo system [45,46] has been considered in Volpert et al [34]. A piecewise linear reaction function has been used [47,48].…”

Section: (V) Fronts In Multi-component Systemsmentioning

confidence: 99%

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Fronts in anomalous diffusion–reaction systems

Volpert

Nec

Nepomnyashchy

2013

Phil. Trans. R. Soc. A.

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A review of recent developments in the field of front dynamics in anomalous diffusion–reaction systems is presented. Both fronts between stable phases and those propagating into an unstable phase are considered. A number of models of anomalous diffusion with reaction are discussed, including models with Lévy flights, truncated Lévy flights, subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators.

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Section: (I) Travelling Wave Solutionsmentioning

confidence: 99%

“…The propagation of fronts in a bistable superdiffusive system governed by the asymmetric model (2.6) with the reaction function (2.3) has been considered in recent studies [34,41].…”

Section: (I) Travelling Wave Solutionsmentioning

confidence: 99%

Section: (Iii) Pinning Of Frontsmentioning

confidence: 99%

“…A superdiffusive version of the FitzHugh-Nagumo system [45,46] has been considered in Volpert et al [34]. A piecewise linear reaction function has been used [47,48].…”

Section: (V) Fronts In Multi-component Systemsmentioning

confidence: 99%

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Fronts in anomalous diffusion–reaction systems

Volpert

Nec

Nepomnyashchy

2013

Phil. Trans. R. Soc. A.

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“…[8], where it was shown that the truncation of the Lévy flights due to tempering leads to a transient front acceleration after which the front asymptotically reaches a terminal speed. Other works on anomalous transport in front propagation include studies on: bistable reaction processes and anomalous diffusion caused by Lévy flights [25]; analytic solutions of fractional reaction-diffusion equations [21]; reaction-diffusion systems with bistable reaction terms and directional anomalous diffusion [15]; construction of reaction-sub-diffusion equations [22]; fractional reproduction-dispersal equations and heavy tail dispersal kernels [1]; role of fluctuations in reaction-super-diffusion dynamics [2]; non-Markovian random walks and sub-diffusion in reaction-diffusion systems [13]; exact super-diffusive front propagation solutions with piecewise linear reaction kinetics functions [24]; and front dynamics in two-species competition models driven by Lévy flights [14], among others. The recent work in Ref.…”

Section: Introductionmentioning

confidence: 99%

Front propagation in reaction-diffusion systems with anomalous diffusion

del‐Castillo‐Negrete

2014

Bol. Soc. Mat. Mex.

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A numerical study of the role of anomalous diffusion in front propagation in reaction-diffusion systems is presented. Three models of anomalous diffusion are considered: fractional diffusion, tempered fractional diffusion, and a model that combines fractional diffusion and regular diffusion. The reaction kinetics corresponds to a Fisher-Kolmogorov nonlinearity. The numerical method is based on a finite-difference operator splitting algorithm with an explicit Euler step for the time advance of the reaction kinetics, and a Crank-Nicholson semi-implicit time step for the transport operator. The anomalous diffusion operators are discretized using an upwind, flux-conserving, Grunwald-Letnikov finite-difference scheme applied to the regularized fractional derivatives. With fractional diffusion of order $\alpha$, fronts exhibit exponential acceleration, $a_L(t) \sim e^{\gamma t/\alpha}$, and develop algebraic decaying tails, $\phi \sim 1/x^{\alpha}$. In the case of tempered fractional diffusion, this phenomenology prevails in the intermediate asymptotic regime $\left(\chi t \right)^{1/\alpha} \ll x \ll 1/\lambda$, where $1/\lambda$ is the scale of the tempering. Outside this regime, i.e. for $x > 1/\lambda$, the tail exhibits the tempered decay $\phi \sim e^{-\lambda x}/x^{\alpha+1}$, and the front velocity approaches the terminal speed $v_*= \left(\gamma-\lambda^\alpha \chi\right)/ \lambda$. Of particular interest is the study of the interplay of regular and fractional diffusion. It is shown that the main role of regular diffusion is to delay the onset of front acceleration. In particular, the crossover time, $t_c$, to transition to the accelerated fractional regime exhibits a logarithmic scaling of the form $t_c \sim \log \left(\chi_d/\chi_f\right)$ where $\chi_d$ and $\chi_f$ are the regular and fractional diffusivities

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