Fractional variational calculus for non-differentiable functions,   chapter
  The fractional calculus of variations is now a subject under strong research. Different definitions for fractional derivatives and integrals are used, depending on the purpose under study. In this paper the fractional operators are defined in the sense of Jumarie. This allows us to work with functions which are non-differentiable. We present necessary and sufficient optimality conditions for fractional problems of the calculus of variations with a Lagrangian density depending on the free end-points. 
Agnieszka B. Malinowska
math.OC