On a nonlocal diffusion model with Neumann boundary conditions

Bogoya, Mauricio; Gomez, A S Cesar

doi:10.1016/j.na.2011.12.019

Cited by 17 publications

(13 citation statements)

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“…The nonlinear problem (1) (or (5)) is important for modeling a variety of phenomena of life sciences, including in biology, biochemistry, economics, medicine and physics. Particular cases of the nonlinear second-order boundary value problem (1), supplied with different boundary conditions, have been successfully applied to many complex moving interface problems, e.g., the motion of anti-phase boundaries in crystalline solids [2], the mixture of two incompressible fluids, the nucleation of solids, and vesicle membranes (see [3][4][5] and the references therein). In addition, the nonlinear problems of type (1) 1 , occur in the phase-field transition system (e.g., [6]) where the phase function v(t, x) describes the transition between the solid and liquid phases in the solidification process of a material occupying a region Ω.…”

Section: Introductionmentioning

confidence: 99%

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Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

Moroşanu

Pavăl

2021

Mathematics

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In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) and v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(Q), facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.

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Section: Introductionmentioning

confidence: 99%

“…The Leray-Schauder principle (see [1,4,[11][12][13][14][15][19][20][21] and reference therein); • The L p -theory of linear and quasi-linear parabolic equations; • Green's first identity…”

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confidence: 99%

Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

Moroşanu

Pavăl

2021

Mathematics

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“…The nonlocal diffusion problems have been used to modeling several problems of the applied mathematics, associate with several branch of sciences and engineering, see for instance [1], [2], [3] and references therein. Analytical studies of that models can be found in a lot of references on the theme, however, the following (including its bibliography) can be considered illustrative [4], [5], [6], [7], [8]. A physical description of Eq.…”

Section: Introductionmentioning

confidence: 99%

“…have been studied by the authors in [5]. Results on existence and uniqueness of solutions as well as a comparison principle and others important characteristics of (2) was obtained.…”

Section: Introductionmentioning

confidence: 99%

On stationary solutions and blow-up for a discrete model with nonlocal diffusion

Gómez¹,

Bogoya²

2018

ces

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In this paper, we study some results relatives to a nonlocal diffusion model with Neumann boundary conditions in the discrete case. In particular, aspects relatives to stationary solutions are derived. We also study the blow-up phenomenon of the solutions and we find the blow up rates when the boundary conditions is given by the function u p with p > 1. The asymptotic behavior of the solutions as t goes to infinity is studied. Finally, we show some numerical experiments which illustrate our results.

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“…Indeed, the problem has been generally faced analytically, investigating the existence and uniqueness of solutions as different boundary conditions are imposed [12]. From a physical point of view, anomalous diffusion is generally related to non-Markovian characteristics of the systems such as fractality, interactions and memory effects [13].…”

Section: Introductionmentioning

confidence: 99%

Diffusion problems on fractional nonlocal media

2013

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Abstract:In this paper, the nonlocal diffusion in one-dimensional continua is investigated by means of a fractional calculus approach. The problem is set on finite spatial domains and it is faced numerically by means of fractional finite differences, both for what concerns the transient and the steady-state regimes. Nonlinear deviations from classical solutions are observed. Furthermore, it is shown that fractional operators possess a clear physical-mechanical meaning, representing conductors, whose conductance decays as a powerlaw of the distance, connecting non-adjacent points of the body.

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On a nonlocal diffusion model with Neumann boundary conditions

Cited by 17 publications

References 25 publications

Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

On stationary solutions and blow-up for a discrete model with nonlocal diffusion

Diffusion problems on fractional nonlocal media

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