Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models

Let J: ℝ → ℝ be a nonnegative, smooth function with ∫ℝ J(r)dr = 1, supported in [-1, 1], symmetric, J(r) = J(-r), and strictly increasing in [-1,0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation ut(x,...

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Autor principal: Rossi, Julio Daniel
Publicado: 2007
Materias:
Neumann boundary conditions
Nonlocal diffusion
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v135_n12_p3837_Bogoya
http://hdl.handle.net/20.500.12110/paper_00029939_v135_n12_p3837_Bogoya
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00029939_v135_n12_p3837_Bogoya
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spelling paper:paper_00029939_v135_n12_p3837_Bogoya2023-06-08T14:23:28Z Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models Rossi, Julio Daniel Neumann boundary conditions Nonlocal diffusion Let J: ℝ → ℝ be a nonnegative, smooth function with ∫ℝ J(r)dr = 1, supported in [-1, 1], symmetric, J(r) = J(-r), and strictly increasing in [-1,0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation ut(x, t)=∫L-L(J(x-y/ u(y,t) - J(x-y/u(x, t))dy, x∈[-L, L].We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as t → ∞: they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments. © 2007 American Mathematical Society. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2007 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v135_n12_p3837_Bogoya http://hdl.handle.net/20.500.12110/paper_00029939_v135_n12_p3837_Bogoya
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Neumann boundary conditions
Nonlocal diffusion
spellingShingle Neumann boundary conditions
Nonlocal diffusion
Rossi, Julio Daniel
Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models
topic_facet Neumann boundary conditions
Nonlocal diffusion
description Let J: ℝ → ℝ be a nonnegative, smooth function with ∫ℝ J(r)dr = 1, supported in [-1, 1], symmetric, J(r) = J(-r), and strictly increasing in [-1,0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation ut(x, t)=∫L-L(J(x-y/ u(y,t) - J(x-y/u(x, t))dy, x∈[-L, L].We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as t → ∞: they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments. © 2007 American Mathematical Society.
author Rossi, Julio Daniel
author_facet Rossi, Julio Daniel
author_sort Rossi, Julio Daniel
title Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models
title_short Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models
title_full Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models
title_fullStr Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models
title_full_unstemmed Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models
title_sort neumann boundary conditions for a nonlocal nonlinear diffusion operator. continuous and discrete models
publishDate 2007
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v135_n12_p3837_Bogoya
http://hdl.handle.net/20.500.12110/paper_00029939_v135_n12_p3837_Bogoya
work_keys_str_mv AT rossijuliodaniel neumannboundaryconditionsforanonlocalnonlineardiffusionoperatorcontinuousanddiscretemodels
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