Euler–Lagrange-Type Equations for Functionals Involving Fractional Operators and Antiderivatives

The goal of this paper is to present the necessary and sufficient conditions that every extremizer of a given class of functionals, defined on the set C1[a,b], must satisfy. The Lagrange function depends on a generalized fractional derivative, on a generalized fractional integral, and on an antideri...

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Bibliographic Details
Published in:Mathematics (Basel) 2023-07, Vol.11 (14), p.3208
Main Author: Almeida, Ricardo
Format: Article
Language:English
Subjects:
Boundary conditions
Calculus
Calculus of variations
Convexity
Derivatives
Euler-Lagrange equation
Fractional calculus
generalized fractional derivative
Integrals
Operators (mathematics)
Optimization
Ordinary differential equations
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Summary:The goal of this paper is to present the necessary and sufficient conditions that every extremizer of a given class of functionals, defined on the set C1[a,b], must satisfy. The Lagrange function depends on a generalized fractional derivative, on a generalized fractional integral, and on an antiderivative involving the previous fractional operators. We begin by obtaining the fractional Euler–Lagrange equation, which is a necessary condition to optimize a given functional. By imposing convexity conditions over the Lagrange function, we prove that it is also a sufficient condition for optimization. After this, we consider variational problems with additional constraints on the set of admissible functions, such as the isoperimetric and the holonomic problems. We end by considering a generalization of the fundamental problem, where the fractional order is not restricted to real values between 0 and 1, but may take any positive real value. We also present some examples to illustrate our results.
ISSN:2227-7390
2227-7390
DOI:10.3390/math11143208