Reflection symmetric formulation of generalized fractional variational calculus

Abstract

We define generalized fractional derivatives (GFDs) symmetric and anti-symmetric w.r.t. the reflection symmetry in a finite interval. Arbitrary functions are split into parts with well defined reflection symmetry properties in a hierarchy of intervals [0, b/2 m ], m ε ℕ0. For these parts - [J]-projections of function, we derive the representation formulas for generalized fractional operators (GFOs) and examine integration properties. It appears that GFOs can be reduced to operators determined in subintervals [0, b/2 m ]. The results are applied in the derivation of Euler-Lagrange equations for action dependent on Riemann-Liouville type GFDs. We show that for Lagrangian being a sum (finite or not) of monomials, the obtained equations of motion can be localized in arbitrary short subinterval [0, b/2 m ]. © 2013 Versita Warsaw and Springer-Verlag Wien.