The Euler-Lagrange and Legendre equations for functionals involving distributed-order fractional derivatives

In this paper, we extend some fractional calculus of variations results by considering functionals depending on distributed-order fractional derivatives. Using variational techniques, we deduce optimal necessary conditions of Euler-Lagrange and Legendre type. We also study the case where integral and holonomic constraints are imposed. Finally, a numerical procedure is given to solve some proposed problems.