Evolution problems with local/nonlocal coupling

Detalhes bibliográficos
Ano de defesa: 2021
Autor(a) principal: Bruna Cassol dos Santos
Orientador(a): Sergio Muniz Oliva Filho
Banca de defesa: José María Arrieta Algarra, Fernando Quirós Gracián, Julio Daniel Rossi, Joana Isabel Afonso Mourão Terra
Tipo de documento: Tese
Tipo de acesso: Acesso aberto
Idioma: eng
Instituição de defesa: Universidade de São Paulo
Programa de Pós-Graduação: Matemática Aplicada
Departamento: Não Informado pela instituição
País: BR
Link de acesso: https://doi.org/10.11606/T.45.2021.tde-05082021-085051
Resumo: Classical models, such as Partial Differential Equations (PDE), are widely used for making local approximations even if they have some limitations for capturing long-range effects. On the other side, the modeling of nonlocal effects is getting attention in many applied areas, like ecology, epidemiology, physics, and engineering. The development of a rigorous theoretical and computational framework for nonlocal models is far less developed than its local counterpart. In this work, we propose and study an evolution problem that couple local and nonlocal equations. The local part is classically represented by the Laplacian operator, while the nonlocal part is represented by a diffusion operator with an integrable kernel in convolution form, J(x y). As a first approximation, we study the properties of the model in the one-dimensional case. Results of existence, uniqueness, mass conservation, and asymptotic decay of solutions were verified. Next, we extend these results to higher dimensions. For the one-dimensional case, with the appropriate rescale of the nonlocal kernel, it is possible to recover the heat equation in the whole domain. Next, we continue our analysis of this coupled problem and, taking advantage of the particular coupling structure, we use the Splitting Operator method to provide a different proof of existence and uniqueness. We also develop some numerical experiments to illustrate the obtained theoretical results. Using classical numerical methods for PDE, we check that the solution of the discrete model converges to the mean value of the initial condition (when we assume Neumann type boundary conditions), as we have shown theoretically. Finally, we study the properties of the evolution problem in a thin domain. We consider the limit case when the nonlocal subdomain is narrowed in one direction, making the nonlocal domain concentrates in a set of smaller dimension. In this way, we obtain a model in which the local and nonlocal parts of the problem are defined in subdomains of different dimensions. We also show that the limit problem shares the same properties obtained in the one-dimensional case; existence and uniqueness, mass conservation, comparison, and asymptotic decay of solutions for large times.
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spelling info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesis Evolution problems with local/nonlocal coupling Problemas de evolução com acoplamento local/não local 2021-07-13Sergio Muniz Oliva FilhoJosé María Arrieta AlgarraFernando Quirós GraciánJulio Daniel RossiJoana Isabel Afonso Mourão TerraBruna Cassol dos SantosUniversidade de São PauloMatemática AplicadaUSPBR Asymptotic behavior Comportamento assintótico Difusão não local Domínios finos Equação do calor Heat equation Métodos numéricos Nonlocal diffusion Numerical methods Thin domains Classical models, such as Partial Differential Equations (PDE), are widely used for making local approximations even if they have some limitations for capturing long-range effects. On the other side, the modeling of nonlocal effects is getting attention in many applied areas, like ecology, epidemiology, physics, and engineering. The development of a rigorous theoretical and computational framework for nonlocal models is far less developed than its local counterpart. In this work, we propose and study an evolution problem that couple local and nonlocal equations. The local part is classically represented by the Laplacian operator, while the nonlocal part is represented by a diffusion operator with an integrable kernel in convolution form, J(x y). As a first approximation, we study the properties of the model in the one-dimensional case. Results of existence, uniqueness, mass conservation, and asymptotic decay of solutions were verified. Next, we extend these results to higher dimensions. For the one-dimensional case, with the appropriate rescale of the nonlocal kernel, it is possible to recover the heat equation in the whole domain. Next, we continue our analysis of this coupled problem and, taking advantage of the particular coupling structure, we use the Splitting Operator method to provide a different proof of existence and uniqueness. We also develop some numerical experiments to illustrate the obtained theoretical results. Using classical numerical methods for PDE, we check that the solution of the discrete model converges to the mean value of the initial condition (when we assume Neumann type boundary conditions), as we have shown theoretically. Finally, we study the properties of the evolution problem in a thin domain. We consider the limit case when the nonlocal subdomain is narrowed in one direction, making the nonlocal domain concentrates in a set of smaller dimension. In this way, we obtain a model in which the local and nonlocal parts of the problem are defined in subdomains of different dimensions. We also show that the limit problem shares the same properties obtained in the one-dimensional case; existence and uniqueness, mass conservation, comparison, and asymptotic decay of solutions for large times. Modelos clássicos, como Equações Diferenciais Parciais (EDPs), são amplamente usados para fazer aproximações locais, mesmo tendo algumas limitações para capturar efeitos de longo alcance. Por outro lado, a modelagem de efeitos não locais está recebendo atenção em muitas áreas aplicadas, como ecologia, epidemiologia, física e engenharia. O desenvolvimento de uma estrutura teórica e computacional rigorosa para modelos não locais ainda está em desenvolvimento em contrapartida a teoria local. Neste trabalho, propomos e estudamos um problema de evolução que acopla equações locais e não locais. A parte local é classicamente representada pelo operador Laplaciano, enquanto a parte não local é representada pelo operador de difusão com um núcleo integrável em forma de convolução, J(x y). Como uma primeira aproximação, estudamos as propriedades do modelo no caso unidimensional. Resultados de existência, unicidade, conservação de massa e decaimento assintótico das soluções foram verificados. A seguir, estendemos esses resultados para dimensões mais altas. Para o caso unidimensional, com o reescalonamento adequado do núcleo não local, é possível recuperar a equação do calor em todo o domínio. Em seguida, continuando nossa análise e, aproveitando as vantagens da estrutura de acoplamento particular, usamos o método Operador de Divisão para fornecer uma prova diferente de existência e unicidade de soluções. Além disso, desenvolvemos alguns experimentos numéricos para ilustrar os resultados teóricos obtidos. Usando métodos numéricos clássicos para EDPs, verificamos que a solução do modelo discreto converge para o valor médio da condição inicial (quando assumimos condições de contorno do tipo Neumann), como mostramos teoricamente. Finalmente, estudamos as propriedades do problema de evolução em um domínio fino. Consideramos o caso limite quando o subdomínio não local é estreitado em uma direção, fazendo com que o domínio não local se concentre em um conjunto de dimensão mais baixa. Dessa forma, obtemos um modelo no qual as partes locais e não locais do problema são definidas em subdomínios de dimensões distintas. Também mostramos que o problema limite compartilha as mesmas propriedades obtidas no caso unidimensional; existência e unicidade, conservação de massa, comparação e decaimento assintótico de soluções, para t suficientemente grande. https://doi.org/10.11606/T.45.2021.tde-05082021-085051info:eu-repo/semantics/openAccessengreponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USP2023-12-21T18:16:29Zoai:teses.usp.br:tde-05082021-085051Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212022-01-28T18:46:02Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false
dc.title.en.fl_str_mv Evolution problems with local/nonlocal coupling
dc.title.alternative.pt.fl_str_mv Problemas de evolução com acoplamento local/não local
title Evolution problems with local/nonlocal coupling
spellingShingle Evolution problems with local/nonlocal coupling
Bruna Cassol dos Santos
title_short Evolution problems with local/nonlocal coupling
title_full Evolution problems with local/nonlocal coupling
title_fullStr Evolution problems with local/nonlocal coupling
title_full_unstemmed Evolution problems with local/nonlocal coupling
title_sort Evolution problems with local/nonlocal coupling
author Bruna Cassol dos Santos
author_facet Bruna Cassol dos Santos
author_role author
dc.contributor.advisor1.fl_str_mv Sergio Muniz Oliva Filho
dc.contributor.referee1.fl_str_mv José María Arrieta Algarra
dc.contributor.referee2.fl_str_mv Fernando Quirós Gracián
dc.contributor.referee3.fl_str_mv Julio Daniel Rossi
dc.contributor.referee4.fl_str_mv Joana Isabel Afonso Mourão Terra
dc.contributor.author.fl_str_mv Bruna Cassol dos Santos
contributor_str_mv Sergio Muniz Oliva Filho
José María Arrieta Algarra
Fernando Quirós Gracián
Julio Daniel Rossi
Joana Isabel Afonso Mourão Terra
description Classical models, such as Partial Differential Equations (PDE), are widely used for making local approximations even if they have some limitations for capturing long-range effects. On the other side, the modeling of nonlocal effects is getting attention in many applied areas, like ecology, epidemiology, physics, and engineering. The development of a rigorous theoretical and computational framework for nonlocal models is far less developed than its local counterpart. In this work, we propose and study an evolution problem that couple local and nonlocal equations. The local part is classically represented by the Laplacian operator, while the nonlocal part is represented by a diffusion operator with an integrable kernel in convolution form, J(x y). As a first approximation, we study the properties of the model in the one-dimensional case. Results of existence, uniqueness, mass conservation, and asymptotic decay of solutions were verified. Next, we extend these results to higher dimensions. For the one-dimensional case, with the appropriate rescale of the nonlocal kernel, it is possible to recover the heat equation in the whole domain. Next, we continue our analysis of this coupled problem and, taking advantage of the particular coupling structure, we use the Splitting Operator method to provide a different proof of existence and uniqueness. We also develop some numerical experiments to illustrate the obtained theoretical results. Using classical numerical methods for PDE, we check that the solution of the discrete model converges to the mean value of the initial condition (when we assume Neumann type boundary conditions), as we have shown theoretically. Finally, we study the properties of the evolution problem in a thin domain. We consider the limit case when the nonlocal subdomain is narrowed in one direction, making the nonlocal domain concentrates in a set of smaller dimension. In this way, we obtain a model in which the local and nonlocal parts of the problem are defined in subdomains of different dimensions. We also show that the limit problem shares the same properties obtained in the one-dimensional case; existence and uniqueness, mass conservation, comparison, and asymptotic decay of solutions for large times.
publishDate 2021
dc.date.issued.fl_str_mv 2021-07-13
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/doctoralThesis
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dc.identifier.uri.fl_str_mv https://doi.org/10.11606/T.45.2021.tde-05082021-085051
url https://doi.org/10.11606/T.45.2021.tde-05082021-085051
dc.language.iso.fl_str_mv eng
language eng
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
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dc.publisher.none.fl_str_mv Universidade de São Paulo
dc.publisher.program.fl_str_mv Matemática Aplicada
dc.publisher.initials.fl_str_mv USP
dc.publisher.country.fl_str_mv BR
publisher.none.fl_str_mv Universidade de São Paulo
dc.source.none.fl_str_mv reponame:Biblioteca Digital de Teses e Dissertações da USP
instname:Universidade de São Paulo (USP)
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reponame_str Biblioteca Digital de Teses e Dissertações da USP
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repository.name.fl_str_mv Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)
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