Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives
Autor(a) principal: | |
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Data de Publicação: | 2017 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/18083 |
Resumo: | In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}:= {}^C\!D_{x_0^+}^{1+\alpha} +{}^C\!D_{y_0^+}^{1+\beta} +{}^C\!D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$ and the fractional derivatives ${}^C\!D_{x_0^+}^{1+\alpha}$, ${}^C\!D_{y_0^+}^{1+\beta}$, ${}^C\!D_{z_0^+}^{1+\gamma}$ are in the Caputo sense. Applying integral transform methods we describe a complete family of eigenfunctions and fundamental solutions of the operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag-Leffler function. From the family of fundamental solutions obtained we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper. |
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Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivativesFractional partial differential equationsFractional Laplace and Dirac operatorsCaputo derivativeEigenfunctionsFundamental solutionIn this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}:= {}^C\!D_{x_0^+}^{1+\alpha} +{}^C\!D_{y_0^+}^{1+\beta} +{}^C\!D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$ and the fractional derivatives ${}^C\!D_{x_0^+}^{1+\alpha}$, ${}^C\!D_{y_0^+}^{1+\beta}$, ${}^C\!D_{z_0^+}^{1+\gamma}$ are in the Caputo sense. Applying integral transform methods we describe a complete family of eigenfunctions and fundamental solutions of the operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag-Leffler function. From the family of fundamental solutions obtained we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper.Taylor & Francis2018-07-20T14:01:01Z2017-06-01T00:00:00Z2017-06-012018-06-01T14:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/18083eng1747-693310.1080/17476933.2016.1250401Ferreira, M.Vieira, N.info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T11:33:58Zoai:ria.ua.pt:10773/18083Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:52:47.284577Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives |
title |
Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives |
spellingShingle |
Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives Ferreira, M. Fractional partial differential equations Fractional Laplace and Dirac operators Caputo derivative Eigenfunctions Fundamental solution |
title_short |
Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives |
title_full |
Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives |
title_fullStr |
Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives |
title_full_unstemmed |
Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives |
title_sort |
Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives |
author |
Ferreira, M. |
author_facet |
Ferreira, M. Vieira, N. |
author_role |
author |
author2 |
Vieira, N. |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Ferreira, M. Vieira, N. |
dc.subject.por.fl_str_mv |
Fractional partial differential equations Fractional Laplace and Dirac operators Caputo derivative Eigenfunctions Fundamental solution |
topic |
Fractional partial differential equations Fractional Laplace and Dirac operators Caputo derivative Eigenfunctions Fundamental solution |
description |
In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}:= {}^C\!D_{x_0^+}^{1+\alpha} +{}^C\!D_{y_0^+}^{1+\beta} +{}^C\!D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$ and the fractional derivatives ${}^C\!D_{x_0^+}^{1+\alpha}$, ${}^C\!D_{y_0^+}^{1+\beta}$, ${}^C\!D_{z_0^+}^{1+\gamma}$ are in the Caputo sense. Applying integral transform methods we describe a complete family of eigenfunctions and fundamental solutions of the operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag-Leffler function. From the family of fundamental solutions obtained we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper. |
publishDate |
2017 |
dc.date.none.fl_str_mv |
2017-06-01T00:00:00Z 2017-06-01 2018-07-20T14:01:01Z 2018-06-01T14:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10773/18083 |
url |
http://hdl.handle.net/10773/18083 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1747-6933 10.1080/17476933.2016.1250401 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Taylor & Francis |
publisher.none.fl_str_mv |
Taylor & Francis |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
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1799137578117496832 |