Euler–Lagrange equations for variational problems involving the Riesz–Hilfer fractional derivative

Abstract

In this paper, we obtain the Euler-Lagrange equations for different kind of variational problems with the Lagrangian function containing the Riesz-Hilfer fractional derivative. Since the Riesz-Hilfer fractional derivative is a generalization for the Riesz-Riemann-Liouville and the Riesz-Caputo derivative, then our results generalize many recent works in which the Lagrangian function involving the Riesz-Riemann-Liouville or the Riesz-Caputo derivative. We also study the problem in the presence of delay derivatives and establish a version for Noether theorem in the Riesz-Hilfer sense. In order to achieve our aims we derive some formulas to integration by parts for the Riesz-Hilfer fractional derivative. In the last section, examples are given to clarify the possibility of applicability of our results. In order to clarify the significant conclusions of the paper, we refer to our techniques enable to study many different variational problems containing the Riesz-Hilfer derivative.