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Fractal Fract
Volume 5
Issue 1
10.3390/fractalfract5010024
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Article

A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter

by
Ricardo Almeida
*,† and
Natália Martins
†
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Fractal Fract. 2021, 5(1), 24; https://doi.org/10.3390/fractalfract5010024
Submission received: 9 February 2021 / Revised: 12 March 2021 / Accepted: 15 March 2021 / Published: 18 March 2021
(This article belongs to the Section General Mathematics, Analysis)
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Abstract

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.
Keywords: fractional calculus; Euler–Lagrange equation; natural boundary conditions; isoperimetric problems; holonomic-constrained problems fractional calculus; Euler–Lagrange equation; natural boundary conditions; isoperimetric problems; holonomic-constrained problems

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MDPI and ACS Style

Almeida, R.; Martins, N. A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter. Fractal Fract. 2021, 5, 24. https://doi.org/10.3390/fractalfract5010024

AMA Style

Almeida R, Martins N. A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter. Fractal and Fractional. 2021; 5(1):24. https://doi.org/10.3390/fractalfract5010024

Chicago/Turabian Style

Almeida, Ricardo, and Natália Martins. 2021. "A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter" Fractal and Fractional 5, no. 1: 24. https://doi.org/10.3390/fractalfract5010024

APA Style

Almeida, R., & Martins, N. (2021). A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter. Fractal and Fractional, 5(1), 24. https://doi.org/10.3390/fractalfract5010024

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