Fractional Variational Calculus with Classical and Combined Caputo   Derivatives
  We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler-Lagrange equations to the basic and isoperimetric problems are proved, as well as transversality conditions. 
Tatiana Odzijewicz, Agnieszka B. Malinowska, Delfim F. M. Torres
math.OC