Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels
In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-12-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/6/12/731 |
| _version_ | 1797458217951821824 |
|---|---|
| author | Fátima Cruz Ricardo Almeida Natália Martins |
| author_facet | Fátima Cruz Ricardo Almeida Natália Martins |
| author_sort | Fátima Cruz |
| collection | DOAJ |
| description | In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. Since the Herglotz-type variational problem is a generalization of the classical variational problem, our main results generalize several results from the fractional calculus of variations. To illustrate the theoretical developments included in this paper, we provide some examples. |
| first_indexed | 2024-03-09T16:34:53Z |
| format | Article |
| id | doaj.art-04da600760f7469d8b7560563936c4b4 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T16:34:53Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-04da600760f7469d8b7560563936c4b42023-11-24T14:57:42ZengMDPI AGFractal and Fractional2504-31102022-12-0161273110.3390/fractalfract6120731Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth KernelsFátima Cruz0Ricardo Almeida1Natália Martins2Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalCenter for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalCenter for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, PortugalIn this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. Since the Herglotz-type variational problem is a generalization of the classical variational problem, our main results generalize several results from the fractional calculus of variations. To illustrate the theoretical developments included in this paper, we provide some examples.https://www.mdpi.com/2504-3110/6/12/731fractional calculuscalculus of variationsEuler–Lagrange equationsHerglotz fractional variational problemshigher-order derivativestime delay |
| spellingShingle | Fátima Cruz Ricardo Almeida Natália Martins Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels Fractal and Fractional fractional calculus calculus of variations Euler–Lagrange equations Herglotz fractional variational problems higher-order derivatives time delay |
| title | Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels |
| title_full | Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels |
| title_fullStr | Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels |
| title_full_unstemmed | Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels |
| title_short | Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels |
| title_sort | herglotz variational problems involving distributed order fractional derivatives with arbitrary smooth kernels |
| topic | fractional calculus calculus of variations Euler–Lagrange equations Herglotz fractional variational problems higher-order derivatives time delay |
| url | https://www.mdpi.com/2504-3110/6/12/731 |
| work_keys_str_mv | AT fatimacruz herglotzvariationalproblemsinvolvingdistributedorderfractionalderivativeswitharbitrarysmoothkernels AT ricardoalmeida herglotzvariationalproblemsinvolvingdistributedorderfractionalderivativeswitharbitrarysmoothkernels AT nataliamartins herglotzvariationalproblemsinvolvingdistributedorderfractionalderivativeswitharbitrarysmoothkernels |