Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/39334 |
Resumo: | The goal of this paper is to present the necessary and sufficient conditions that every extremizer of a given class of functionals, defined on the set ¹[,], must satisfy. The Lagrange function depends on a generalized fractional derivative, on a generalized fractional integral, and on an antiderivative involving the previous fractional operators. We begin by obtaining the fractional Euler–Lagrange equation, which is a necessary condition to optimize a given functional. By imposing convexity conditions over the Lagrange function, we prove that it is also a sufficient condition for optimization. After this, we consider variational problems with additional constraints on the set of admissible functions, such as the isoperimetric and the holonomic problems. We end by considering a generalization of the fundamental problem, where the fractional order is not restricted to real values between 0 and 1, but may take any positive real value. We also present some examples to illustrate our results. |
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Euler–Lagrange-type equations for functionals involving fractional operators and antiderivativesFractional calculusCalculus of variationsGeneralized fractional derivativeThe goal of this paper is to present the necessary and sufficient conditions that every extremizer of a given class of functionals, defined on the set ¹[,], must satisfy. The Lagrange function depends on a generalized fractional derivative, on a generalized fractional integral, and on an antiderivative involving the previous fractional operators. We begin by obtaining the fractional Euler–Lagrange equation, which is a necessary condition to optimize a given functional. By imposing convexity conditions over the Lagrange function, we prove that it is also a sufficient condition for optimization. After this, we consider variational problems with additional constraints on the set of admissible functions, such as the isoperimetric and the holonomic problems. We end by considering a generalization of the fundamental problem, where the fractional order is not restricted to real values between 0 and 1, but may take any positive real value. We also present some examples to illustrate our results.MDPI2023-09-07T15:17:40Z2023-07-02T00:00:00Z2023-07-02info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/39334eng10.3390/math11143208Almeida, Ricardoinfo: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:RCAAP2024-02-22T12:16:56Zoai:ria.ua.pt:10773/39334Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T03:09:34.869862Repositório 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 |
Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives |
title |
Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives |
spellingShingle |
Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives Almeida, Ricardo Fractional calculus Calculus of variations Generalized fractional derivative |
title_short |
Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives |
title_full |
Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives |
title_fullStr |
Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives |
title_full_unstemmed |
Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives |
title_sort |
Euler–Lagrange-type equations for functionals involving fractional operators and antiderivatives |
author |
Almeida, Ricardo |
author_facet |
Almeida, Ricardo |
author_role |
author |
dc.contributor.author.fl_str_mv |
Almeida, Ricardo |
dc.subject.por.fl_str_mv |
Fractional calculus Calculus of variations Generalized fractional derivative |
topic |
Fractional calculus Calculus of variations Generalized fractional derivative |
description |
The goal of this paper is to present the necessary and sufficient conditions that every extremizer of a given class of functionals, defined on the set ¹[,], must satisfy. The Lagrange function depends on a generalized fractional derivative, on a generalized fractional integral, and on an antiderivative involving the previous fractional operators. We begin by obtaining the fractional Euler–Lagrange equation, which is a necessary condition to optimize a given functional. By imposing convexity conditions over the Lagrange function, we prove that it is also a sufficient condition for optimization. After this, we consider variational problems with additional constraints on the set of admissible functions, such as the isoperimetric and the holonomic problems. We end by considering a generalization of the fundamental problem, where the fractional order is not restricted to real values between 0 and 1, but may take any positive real value. We also present some examples to illustrate our results. |
publishDate |
2023 |
dc.date.none.fl_str_mv |
2023-09-07T15:17:40Z 2023-07-02T00:00:00Z 2023-07-02 |
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/10773/39334 |
url |
http://hdl.handle.net/10773/39334 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.3390/math11143208 |
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 |
MDPI |
publisher.none.fl_str_mv |
MDPI |
dc.source.none.fl_str_mv |
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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) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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