  We study two generalizations of fractional variational problems by
considering higher-order derivatives and a state time delay. We prove a
higher-order integration by parts formula involving a Caputo fractional
derivative of variable order and we establish several necessary optimality
conditions for functionals containing a combined Caputo derivative of variable
fractional order. Because the endpoint is considered to be free, we also deduce
associated transversality conditions. In the end, we consider functionals with
a time delay and deduce corresponding optimality conditions. Some examples are
given to illustrate the new results. Computational aspects are discussed using
the open source software package Chebfun.
