How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

By Carmen Cortázar, Manuel Elgueta, Julio D. Rossi, Noemi Wolanski. 2006

202 citations recorded by the source corpus.

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a9f397eea8029037db9789a34354aee1f5562a46

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In Applied Mathematics

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Citations in the corpus

Nonlocal problems with Neumann boundary conditions (2014, background): “5) Other Neumann problems for the fractional Laplacian (or other nonlocal operators) were introduced in [4, 8], [1, 3], [9, 10, 11], and [15].”
On the principal eigenvalue of some nonlocal diffusion problems (2009, background): “This is indeed the case at least for some nonlinearities, even for the parabolic version, see [15], [16], [17], [20].”
Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations (2014, background): “9), Theorem A has been proved in [8] and [9].”
Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term (2009, methodology): “Non-local evolution equations of the form ut (x, t) = ∫ RN J (x−y)u(y, t) dy−u(x, t), and their variations, have been widely used to model diffusion processes, see [1–4,6–8,10–14,18].”
The Neumann problem for nonlocal nonlinear diffusion equations (2008, methodology): “In [10] and [12] the asymptotic behaviour for the linear model is studied and in [ 13 ] it is proved that solutions to the linear problem converge to the solution of the classical heat equation with Neumann boundary conditions when the convolution kernel J is rescaled in a suitable way.”
A logistic equation with refuge and nonlocal diffusion (2009, background): “the “Neumann” boundary condition for the same problem is treated in [1], [13] and [14].”
Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications (2012, background): “The reader is referred to [6,7], and [29] about the approximations of the initial value problems of the random dispersal operators associated to (1.”
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions (2008, background): “For p = 2, in [20] (see also [19]) it is proved that solutions to the linear problem P J (u0) converge to the 2”
On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, background): “If the behaviour of λp(LΩ + a) with respect to a or Ω can be exhibited directly from the definition, the impact of scalings of the kernel is usually unknown and has been largely ignored in the literature except in some specific situations involving particular nonlocal dispersal operators defined in ”
On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications (2013, unknown): “The reader is referred to [1], [2], [4], [5], [6], [7], [8], [10], [11], [16], [22], [23], [24], [26], [28], [30], [32], etc.”