Green's functions for sturm-liouville problems on directed tree graphs

Let $\Gamma$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $\mathcal{L}[u]=(-pu')'+qu$ acting on functions that are continuous on all of $\Gamma$, and twice continuously differentiable in the interior of each edge. The functions $p$ and $q$ a...

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Autores:
Ramirez, Jorge M.
Tipo de recurso:
Article of journal
Fecha de publicación:
2012
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
spa
OAI Identifier:
oai:repositorio.unal.edu.co:unal/42249
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/42249
http://bdigital.unal.edu.co/32346/
Palabra clave:
Problema Sturm-Liouville en grafo
función de Green
34B24
35R02
35J08
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openAccess
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Atribución-NoComercial 4.0 Internacional
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spelling Atribución-NoComercial 4.0 InternacionalDerechos reservados - Universidad Nacional de Colombiahttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Ramirez, Jorge M.ebd11590-841b-43a7-aa47-7332ebd1e1f43002019-06-28T10:39:21Z2019-06-28T10:39:21Z2012https://repositorio.unal.edu.co/handle/unal/42249http://bdigital.unal.edu.co/32346/Let $\Gamma$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $\mathcal{L}[u]=(-pu')'+qu$ acting on functions that are continuous on all of $\Gamma$, and twice continuously differentiable in the interior of each edge. The functions $p$ and $q$ are assumed continuous on each edge, and $p$ strictly positive on $\Gamma$. The problem is to find a solution $f:\Gamma \to \mathbb{R}$ to the problem $\mathcal{L}[f] = h$ with $2m$ additional conditions at the nodes of $\Gamma$. These node conditions include continuity at internal nodes, and jump conditions on the derivatives of $f$ with respect to a positive measure $\rho$. Node conditions are given in the form of linear functionals $\l_1,\ldots,\l_{2m}$ acting on the space of admissible functions. A novel formula is given for the Green's function $G:\Gamma\times \Gamma \to \mathbb{R}$ associated to this problem. Namely, the solution to the semi-homogenous problem $\mathcal{L}[f] = h$, $\l_i[f] =0$ for $i=1,\ldots,2m$ is given by $f(x) = \int_\Gamma G(x,y) h(y)\,d\rho$.application/pdfspaUniversidad Nacuional de Colombia; Sociedad Colombiana de matemáticashttp://revistas.unal.edu.co/index.php/recolma/article/view/31839Universidad Nacional de Colombia Revistas electrónicas UN Revista Colombiana de MatemáticasRevista Colombiana de MatemáticasRevista Colombiana de Matemáticas; Vol. 46, núm. 1 (2012); 15-25 0034-7426Ramirez, Jorge M. (2012) Green's functions for sturm-liouville problems on directed tree graphs. Revista Colombiana de Matemáticas; Vol. 46, núm. 1 (2012); 15-25 0034-7426 .Green's functions for sturm-liouville problems on directed tree graphsArtículo de revistainfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1http://purl.org/coar/version/c_970fb48d4fbd8a85Texthttp://purl.org/redcol/resource_type/ARTProblema Sturm-Liouville en grafofunción de Green34B2435R0235J08ORIGINAL31839-116348-1-PB.pdfapplication/pdf463717https://repositorio.unal.edu.co/bitstream/unal/42249/1/31839-116348-1-PB.pdf2448ee62a718fdfe1cbeab7516dd3e0fMD5131839-142397-1-PB.htmltext/html5977https://repositorio.unal.edu.co/bitstream/unal/42249/2/31839-142397-1-PB.htmlf453d5ccac3bc9233a0412fd25c38594MD52THUMBNAIL31839-116348-1-PB.pdf.jpg31839-116348-1-PB.pdf.jpgGenerated Thumbnailimage/jpeg5117https://repositorio.unal.edu.co/bitstream/unal/42249/3/31839-116348-1-PB.pdf.jpg407f596a5b8314aac684ae41da56b142MD53unal/42249oai:repositorio.unal.edu.co:unal/422492023-02-06 23:16:19.61Repositorio Institucional Universidad Nacional de Colombiarepositorio_nal@unal.edu.co
dc.title.spa.fl_str_mv Green's functions for sturm-liouville problems on directed tree graphs
title Green's functions for sturm-liouville problems on directed tree graphs
spellingShingle Green's functions for sturm-liouville problems on directed tree graphs
Problema Sturm-Liouville en grafo
función de Green
34B24
35R02
35J08
title_short Green's functions for sturm-liouville problems on directed tree graphs
title_full Green's functions for sturm-liouville problems on directed tree graphs
title_fullStr Green's functions for sturm-liouville problems on directed tree graphs
title_full_unstemmed Green's functions for sturm-liouville problems on directed tree graphs
title_sort Green's functions for sturm-liouville problems on directed tree graphs
dc.creator.fl_str_mv Ramirez, Jorge M.
dc.contributor.author.spa.fl_str_mv Ramirez, Jorge M.
dc.subject.proposal.spa.fl_str_mv Problema Sturm-Liouville en grafo
función de Green
34B24
35R02
35J08
topic Problema Sturm-Liouville en grafo
función de Green
34B24
35R02
35J08
description Let $\Gamma$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $\mathcal{L}[u]=(-pu')'+qu$ acting on functions that are continuous on all of $\Gamma$, and twice continuously differentiable in the interior of each edge. The functions $p$ and $q$ are assumed continuous on each edge, and $p$ strictly positive on $\Gamma$. The problem is to find a solution $f:\Gamma \to \mathbb{R}$ to the problem $\mathcal{L}[f] = h$ with $2m$ additional conditions at the nodes of $\Gamma$. These node conditions include continuity at internal nodes, and jump conditions on the derivatives of $f$ with respect to a positive measure $\rho$. Node conditions are given in the form of linear functionals $\l_1,\ldots,\l_{2m}$ acting on the space of admissible functions. A novel formula is given for the Green's function $G:\Gamma\times \Gamma \to \mathbb{R}$ associated to this problem. Namely, the solution to the semi-homogenous problem $\mathcal{L}[f] = h$, $\l_i[f] =0$ for $i=1,\ldots,2m$ is given by $f(x) = \int_\Gamma G(x,y) h(y)\,d\rho$.
publishDate 2012
dc.date.issued.spa.fl_str_mv 2012
dc.date.accessioned.spa.fl_str_mv 2019-06-28T10:39:21Z
dc.date.available.spa.fl_str_mv 2019-06-28T10:39:21Z
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dc.relation.ispartof.spa.fl_str_mv Universidad Nacional de Colombia Revistas electrónicas UN Revista Colombiana de Matemáticas
Revista Colombiana de Matemáticas
dc.relation.ispartofseries.none.fl_str_mv Revista Colombiana de Matemáticas; Vol. 46, núm. 1 (2012); 15-25 0034-7426
dc.relation.references.spa.fl_str_mv Ramirez, Jorge M. (2012) Green's functions for sturm-liouville problems on directed tree graphs. Revista Colombiana de Matemáticas; Vol. 46, núm. 1 (2012); 15-25 0034-7426 .
dc.rights.spa.fl_str_mv Derechos reservados - Universidad Nacional de Colombia
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dc.rights.license.spa.fl_str_mv Atribución-NoComercial 4.0 Internacional
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Derechos reservados - Universidad Nacional de Colombia
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