A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter
In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-03-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/5/1/24 |
| _version_ | 1797540843440046080 |
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| author | Ricardo Almeida Natália Martins |
| author_facet | Ricardo Almeida Natália Martins |
| author_sort | Ricardo Almeida |
| collection | DOAJ |
| description | In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results. |
| first_indexed | 2024-03-10T13:06:48Z |
| format | Article |
| id | doaj.art-6e2a47e3ac1c43939f7d2883325e19fa |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T13:06:48Z |
| publishDate | 2021-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-6e2a47e3ac1c43939f7d2883325e19fa2023-11-21T11:05:13ZengMDPI AGFractal and Fractional2504-31102021-03-01512410.3390/fractalfract5010024A Generalization of a Fractional Variational Problem with Dependence on the Boundaries 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, PortugalIn this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results.https://www.mdpi.com/2504-3110/5/1/24fractional calculusEuler–Lagrange equationnatural boundary conditionsisoperimetric problemsholonomic-constrained problems |
| spellingShingle | Ricardo Almeida Natália Martins A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter Fractal and Fractional fractional calculus Euler–Lagrange equation natural boundary conditions isoperimetric problems holonomic-constrained problems |
| title | A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter |
| title_full | A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter |
| title_fullStr | A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter |
| title_full_unstemmed | A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter |
| title_short | A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter |
| title_sort | generalization of a fractional variational problem with dependence on the boundaries and a real parameter |
| topic | fractional calculus Euler–Lagrange equation natural boundary conditions isoperimetric problems holonomic-constrained problems |
| url | https://www.mdpi.com/2504-3110/5/1/24 |
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