Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function
In this paper, we investigate the necessary conditions to optimize a given functional, involving a generalization of the tempered fractional derivative. The exponential function is replaced by the Mittag–Leffler function, and the kernel depends on an arbitrary increasing function. The Lagrangian dep...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/6/477 |
| _version_ | 1797594739706429440 |
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| author | Ricardo Almeida |
| author_facet | Ricardo Almeida |
| author_sort | Ricardo Almeida |
| collection | DOAJ |
| description | In this paper, we investigate the necessary conditions to optimize a given functional, involving a generalization of the tempered fractional derivative. The exponential function is replaced by the Mittag–Leffler function, and the kernel depends on an arbitrary increasing function. The Lagrangian depends on time, the state function, its fractional derivative, and we add a terminal cost function to the formulation of the problem. Since this new fractional derivative is presented in a general form, some previous works are our own particular cases. In addition, for different choices of the kernel, new results can be deduced. Using variational techniques, the fractional Euler–Lagrange equation is proved, as are its associated transversality conditions. The variational problem with additional constraints is also considered. Then, the question of minimizing functionals with an infinite interval of integration is addressed. To end, we study the case of the Herglotz variational problem, which generalizes the previous one. With this work, several optimization conditions are proven that can be useful for different optimization problems dealing with various fractional derivatives. |
| first_indexed | 2024-03-11T02:26:29Z |
| format | Article |
| id | doaj.art-880b49783b644a62b690b6ec9ec0b36d |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T02:26:29Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-880b49783b644a62b690b6ec9ec0b36d2023-11-18T10:29:50ZengMDPI AGFractal and Fractional2504-31102023-06-017647710.3390/fractalfract7060477Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler FunctionRicardo Almeida0Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalIn this paper, we investigate the necessary conditions to optimize a given functional, involving a generalization of the tempered fractional derivative. The exponential function is replaced by the Mittag–Leffler function, and the kernel depends on an arbitrary increasing function. The Lagrangian depends on time, the state function, its fractional derivative, and we add a terminal cost function to the formulation of the problem. Since this new fractional derivative is presented in a general form, some previous works are our own particular cases. In addition, for different choices of the kernel, new results can be deduced. Using variational techniques, the fractional Euler–Lagrange equation is proved, as are its associated transversality conditions. The variational problem with additional constraints is also considered. Then, the question of minimizing functionals with an infinite interval of integration is addressed. To end, we study the case of the Herglotz variational problem, which generalizes the previous one. With this work, several optimization conditions are proven that can be useful for different optimization problems dealing with various fractional derivatives.https://www.mdpi.com/2504-3110/7/6/477fractional calculuscalculus of variationsEuler–Lagrange equationstempered fractional derivativeMittag–Leffler function |
| spellingShingle | Ricardo Almeida Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function Fractal and Fractional fractional calculus calculus of variations Euler–Lagrange equations tempered fractional derivative Mittag–Leffler function |
| title | Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function |
| title_full | Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function |
| title_fullStr | Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function |
| title_full_unstemmed | Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function |
| title_short | Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function |
| title_sort | variational problems involving a generalized fractional derivative with dependence on the mittag leffler function |
| topic | fractional calculus calculus of variations Euler–Lagrange equations tempered fractional derivative Mittag–Leffler function |
| url | https://www.mdpi.com/2504-3110/7/6/477 |
| work_keys_str_mv | AT ricardoalmeida variationalproblemsinvolvingageneralizedfractionalderivativewithdependenceonthemittaglefflerfunction |