Maria Lupa

Reflection symmetric formulation of generalized fractional variational calculus

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/2m], m ∈ 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/2m]. 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/2m].

Reflection Symmetry in Fractional Calculus – Properties and Applications

In this paper we define Riesz type derivatives symmetric and anti-symmetric w.r.t. the reflection mapping in finite interval [a,b]. Functions determined in [a,b] are split into parts with well determined reflection symmetry properties in a hierarchy of intervals [a m ,b m ], m ∈ ℕ, concentrated around an arbitrary point. For these parts - called the [J]-projections of function, we prove the representation and integration formulas for the introduced fractional symmetric and anti-symmetric integrals and derivatives. It appears that they can be reduced to operators determined in arbitrarily short subintervals [a m ,b m ]. The future application in the reflection symmetric fractional variational calculus and the generalization of previous results on localization of Euler-Lagrange equations are discussed.