Métodos numéricos para problemas no locales de evolución
- Autores
- Mastroberti Bersetche, Francisco Vicente
- Año de publicación
- 2019
- Idioma
- inglés
- Tipo de recurso
- tesis doctoral
- Estado
- versión publicada
- Colaborador/a o director/a de tesis
- Acosta Rodriguez, Gabriel
- Descripción
- This work introduces and analyzes a finite element scheme forevolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time weconsider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discusswell-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linearelements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experimentsin one- and two-dimensional domains.
The aim of this work is to study numerical approximations for evolution problems of the form C ∂ α t u + (−∆)su = f in Ω × (0, T), where (−∆)s stands for the fractional Laplacian operator in its integral form and C ∂ α t u(x, t) represents the Caputo derivative. To be more precise, (−∆)su(x) = C(n, s) p.v. ˆ Rn u(x) − u(y) |x − y| n+2s dy, and C ∂ α t u(x, t) = ( 1 Γ(k−α) ´ t 0 1 (t−r)α−k+1 ∂ ku ∂rk (x, r) dr if k − 1 < α < k, k ∈ N, ∂ ku ∂tk u(x, t) if α = k ∈ N. We deal with existence, uniqueness and regularity of solutions in the linear context (i.e. f = f(x, t)). The cases under study include fractional counterparts of the standard diffusion and wave models. Linear finite elements are used for the spatial variable and convolution quadrature techniques for handling the time fractional operator. Error bounds, uniform in the discretization parameters for values of t away from zero, are given. These results are extended to the semi-linear case with f(u) = u − u 3 appearing in the classical Allen-Cahn equations modeling phase separation for binary alloys. Additionally, the asymptotic behaviour of the solutions for s → 0 is studied in this particular context. Implementation details, particularly for the finite element method involving full fractional stiffness matrices and numerical quadratures for singular kernels, are carefully documented
Fil: Mastroberti Bersetche, Francisco Vicente. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina - Materia
-
Laplaciano Fraccionario
Derivada de Caputo
Método de Elementos Finitos - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
- Repositorio
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/80087
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Métodos numéricos para problemas no locales de evoluciónNumerical methods for non-local evolution problemsMastroberti Bersetche, Francisco VicenteLaplaciano FraccionarioDerivada de CaputoMétodo de Elementos Finitoshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1This work introduces and analyzes a finite element scheme forevolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time weconsider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discusswell-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linearelements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experimentsin one- and two-dimensional domains.The aim of this work is to study numerical approximations for evolution problems of the form C ∂ α t u + (−∆)su = f in Ω × (0, T), where (−∆)s stands for the fractional Laplacian operator in its integral form and C ∂ α t u(x, t) represents the Caputo derivative. To be more precise, (−∆)su(x) = C(n, s) p.v. ˆ Rn u(x) − u(y) |x − y| n+2s dy, and C ∂ α t u(x, t) = ( 1 Γ(k−α) ´ t 0 1 (t−r)α−k+1 ∂ ku ∂rk (x, r) dr if k − 1 < α < k, k ∈ N, ∂ ku ∂tk u(x, t) if α = k ∈ N. We deal with existence, uniqueness and regularity of solutions in the linear context (i.e. f = f(x, t)). The cases under study include fractional counterparts of the standard diffusion and wave models. Linear finite elements are used for the spatial variable and convolution quadrature techniques for handling the time fractional operator. Error bounds, uniform in the discretization parameters for values of t away from zero, are given. These results are extended to the semi-linear case with f(u) = u − u 3 appearing in the classical Allen-Cahn equations modeling phase separation for binary alloys. Additionally, the asymptotic behaviour of the solutions for s → 0 is studied in this particular context. Implementation details, particularly for the finite element method involving full fractional stiffness matrices and numerical quadratures for singular kernels, are carefully documentedFil: Mastroberti Bersetche, Francisco Vicente. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaAcosta Rodriguez, Gabriel2019-03-06info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_db06info:ar-repo/semantics/tesisDoctoralapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/80087Mastroberti Bersetche, Francisco Vicente; Acosta Rodriguez, Gabriel; Métodos numéricos para problemas no locales de evolución; 6-3-2019CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://cms.dm.uba.ar/academico/carreras/doctorado/Tesis%20mastroberti.pdfinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:43:58Zoai:ri.conicet.gov.ar:11336/80087instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 09:43:58.789CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
dc.title.none.fl_str_mv |
Métodos numéricos para problemas no locales de evolución Numerical methods for non-local evolution problems |
title |
Métodos numéricos para problemas no locales de evolución |
spellingShingle |
Métodos numéricos para problemas no locales de evolución Mastroberti Bersetche, Francisco Vicente Laplaciano Fraccionario Derivada de Caputo Método de Elementos Finitos |
title_short |
Métodos numéricos para problemas no locales de evolución |
title_full |
Métodos numéricos para problemas no locales de evolución |
title_fullStr |
Métodos numéricos para problemas no locales de evolución |
title_full_unstemmed |
Métodos numéricos para problemas no locales de evolución |
title_sort |
Métodos numéricos para problemas no locales de evolución |
dc.creator.none.fl_str_mv |
Mastroberti Bersetche, Francisco Vicente |
author |
Mastroberti Bersetche, Francisco Vicente |
author_facet |
Mastroberti Bersetche, Francisco Vicente |
author_role |
author |
dc.contributor.none.fl_str_mv |
Acosta Rodriguez, Gabriel |
dc.subject.none.fl_str_mv |
Laplaciano Fraccionario Derivada de Caputo Método de Elementos Finitos |
topic |
Laplaciano Fraccionario Derivada de Caputo Método de Elementos Finitos |
purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
dc.description.none.fl_txt_mv |
This work introduces and analyzes a finite element scheme forevolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time weconsider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discusswell-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linearelements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experimentsin one- and two-dimensional domains. The aim of this work is to study numerical approximations for evolution problems of the form C ∂ α t u + (−∆)su = f in Ω × (0, T), where (−∆)s stands for the fractional Laplacian operator in its integral form and C ∂ α t u(x, t) represents the Caputo derivative. To be more precise, (−∆)su(x) = C(n, s) p.v. ˆ Rn u(x) − u(y) |x − y| n+2s dy, and C ∂ α t u(x, t) = ( 1 Γ(k−α) ´ t 0 1 (t−r)α−k+1 ∂ ku ∂rk (x, r) dr if k − 1 < α < k, k ∈ N, ∂ ku ∂tk u(x, t) if α = k ∈ N. We deal with existence, uniqueness and regularity of solutions in the linear context (i.e. f = f(x, t)). The cases under study include fractional counterparts of the standard diffusion and wave models. Linear finite elements are used for the spatial variable and convolution quadrature techniques for handling the time fractional operator. Error bounds, uniform in the discretization parameters for values of t away from zero, are given. These results are extended to the semi-linear case with f(u) = u − u 3 appearing in the classical Allen-Cahn equations modeling phase separation for binary alloys. Additionally, the asymptotic behaviour of the solutions for s → 0 is studied in this particular context. Implementation details, particularly for the finite element method involving full fractional stiffness matrices and numerical quadratures for singular kernels, are carefully documented Fil: Mastroberti Bersetche, Francisco Vicente. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina |
description |
This work introduces and analyzes a finite element scheme forevolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time weconsider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discusswell-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linearelements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experimentsin one- and two-dimensional domains. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019-03-06 |
dc.type.none.fl_str_mv |
info:eu-repo/semantics/doctoralThesis info:eu-repo/semantics/publishedVersion http://purl.org/coar/resource_type/c_db06 info:ar-repo/semantics/tesisDoctoral |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.none.fl_str_mv |
http://hdl.handle.net/11336/80087 Mastroberti Bersetche, Francisco Vicente; Acosta Rodriguez, Gabriel; Métodos numéricos para problemas no locales de evolución; 6-3-2019 CONICET Digital CONICET |
url |
http://hdl.handle.net/11336/80087 |
identifier_str_mv |
Mastroberti Bersetche, Francisco Vicente; Acosta Rodriguez, Gabriel; Métodos numéricos para problemas no locales de evolución; 6-3-2019 CONICET Digital CONICET |
dc.language.none.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/url/http://cms.dm.uba.ar/academico/carreras/doctorado/Tesis%20mastroberti.pdf |
dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
eu_rights_str_mv |
openAccess |
rights_invalid_str_mv |
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
dc.format.none.fl_str_mv |
application/pdf application/pdf |
dc.source.none.fl_str_mv |
reponame:CONICET Digital (CONICET) instname:Consejo Nacional de Investigaciones Científicas y Técnicas |
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CONICET Digital (CONICET) |
collection |
CONICET Digital (CONICET) |
instname_str |
Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.name.fl_str_mv |
CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.mail.fl_str_mv |
dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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score |
13.13397 |