New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter
This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Capu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-04-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/4/592 |
| _version_ | 1797538953683795968 |
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| author | Ricardo Almeida Natália Martins |
| author_facet | Ricardo Almeida Natália Martins |
| author_sort | Ricardo Almeida |
| collection | DOAJ |
| description | This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided. |
| first_indexed | 2024-03-10T12:38:31Z |
| format | Article |
| id | doaj.art-4626902a561b4abcbab002a02575eecd |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T12:38:31Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-4626902a561b4abcbab002a02575eecd2023-11-21T14:04:06ZengMDPI AGSymmetry2073-89942021-04-0113459210.3390/sym13040592New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real ParameterRicardo Almeida0Natália Martins1Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalCenter for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalThis work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.https://www.mdpi.com/2073-8994/13/4/592fractional calculusEuler–Lagrange equationnatural boundary conditionstime delay |
| spellingShingle | Ricardo Almeida Natália Martins New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter Symmetry fractional calculus Euler–Lagrange equation natural boundary conditions time delay |
| title | New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter |
| title_full | New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter |
| title_fullStr | New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter |
| title_full_unstemmed | New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter |
| title_short | New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter |
| title_sort | new variational problems with an action depending on generalized fractional derivatives the free endpoint conditions and a real parameter |
| topic | fractional calculus Euler–Lagrange equation natural boundary conditions time delay |
| url | https://www.mdpi.com/2073-8994/13/4/592 |
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