Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

Zan, Cecilia De; Soravia, Pierpaolo

doi:10.1016/j.na.2020.111989

Cited by 3 publications

(3 citation statements)

References 17 publications

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“…This property has been proved by different approaches, see e.g. [30] for a probabilistic approach and [5,6,7,15] for a viscosity solution approach. Particularly, in our spatially periodic case, Gärtner's result [30] suggested that, as L → +∞, the interface propagates at a mean speed equal to c * .…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

Bistable pulsating fronts in slowly oscillating environments *

Ding,

Hamel,

Liang

2022

Preprint

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We consider reaction-diffusion fronts in spatially periodic bistable media with large periods. Whereas the homogenization regime associated with small periods had been well studied for bistable or Fisher-KPP reactions and, in the latter case, a formula for the limit minimal speeds of fronts in media with large periods had also been obtained thanks to the linear formulation of these minimal speeds and their monotonicity with respect to the period, the main remaining open question is concerned with fronts in bistable environments with large periods. In bistable media the unique front speeds are not linearly determined and are not monotone with respect to the spatial period in general, making the analysis of the limit of large periods more intricate. We show in this paper the existence of and an explicit formula for the limit of bistable front speeds as the spatial period goes to infinity. We also prove that the front profiles converge to a family of front profiles associated with spatially homogeneous equations. The main results are based on uniform estimates on the spatial width of the fronts, which themselves use zero number properties and intersection arguments.

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Section: Introduction and Main Resultsmentioning

confidence: 93%

Bistable pulsating fronts in slowly oscillating environments *

Ding,

Hamel,

Liang

2022

Preprint

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“…One of the special cases of partial differential equations (PDEs) is reaction diffusion equation (RDE) that has attracted the attention of many researchers, recently [1,20,28,32,33]. RDEs are the mathematical models which correspond with physical and chemical phenomena.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Reaction–Diffusion Equations with Convergence Analysis

Heidari

Ghovatmand

Skandari

et al. 2022

J Nonlinear Math Phys

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In this manuscript, we implement a spectral collocation method to find the solution of the reaction–diffusion equation with some initial and boundary conditions. We approximate the solution of equation by using a two-dimensional interpolating polynomial dependent to the Legendre–Gauss–Lobatto collocation points. We fully show that the achieved approximate solutions are convergent to the exact solution when the number of collocation points increases. We demonstrate the capability and efficiency of the method by providing four numerical examples and comparing them with other available methods.

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“…A. N. Gorban [23] investigates a model reduction in chemical dynamics with slow invariant manifolds and singular perturbations. Bor-Yann Chen, Liying Wu and Junming Hong [24] consider singular limits of reaction diffusion equations and geometric flows with discontinuous velocity.…”

Section: Introductionmentioning

confidence: 99%

The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

Bouatta

Vasilyev

Vinitsky

2021

Discrete and Continuous Models

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The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

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Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

Cited by 3 publications

References 17 publications

Bistable pulsating fronts in slowly oscillating environments *

Bistable pulsating fronts in slowly oscillating environments *

Numerical Solution of Reaction–Diffusion Equations with Convergence Analysis

The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

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