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Derivatives and integrals of non-integer order were introduced more thanthree centuries ago, but only recently gained more attention due to theirapplication on nonlocal phenomena. In this context, the Caputo derivatives arethe most popular approach to fractional calculus among physicists, sincedifferential equations involving Caputo derivatives require regular boundaryconditions. Motivated by several applications in physics and other sciences,the fractional calculus of variations is currently in fast development.However, all current formulations for the fractional variational calculus failto give an Euler-Lagrange equation with only Caputo derivatives. In this work,we propose a new approach to the fractional calculus of variations bygeneralizing the DuBois-Reymond lemma and showing how Euler-Lagrange equationsinvolving only Caputo derivatives can be obtained.

The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo