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

Ricardo Almeida ()
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Ricardo Almeida: Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Mathematics, 2023, vol. 11, issue 14, 1-16

Abstract: 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 C 1 [ 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.

Keywords: fractional calculus; calculus of variations; generalized fractional derivative (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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