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. Arbi- MSC 2010 : Primary 26A33; Secondary 34A08, 49S05, 70H03

A Caputo fractional derivative of a function with respect to another function

“…This is a very rich field, and for it we find several definitions for fractional integrals and for fractional derivatives [16,25]. Just to mention a few, for fractional integrals, given an integrable function f : To overcome the vast number of definitions, we can for instance consider general operators, from which choosing special kernels and some form of differential operator, we obtain the classical fractional integrals and derivatives [1,17,20]. For example, for the kernel k(x, t) = x − t and the differential operator d/dx, we obtain the Riemann-Liouville fractional derivative, and for k(x, t) = ln(x/t) and the differential x d/dx, we obtain the Hadamard fractional derivative.…”

Fractional calculus of variations of several independent variables

“…The notation was introduced in [2] and is now standard [16,25]. Now, we shall define generalized partial fractional operators.…”

Advanced Methods in the Fractional Calculus of Variations

“…In 2010, an interesting perspective to the subject, unifying all mentioned notions of fractional derivatives and integrals, was introduced in Agrawal (2010) and later studied in , Klimek and Lupa (2013), Odzijewicz et al ( , b, 2013a. Precisely, authors considered general operators, which by choosing special kernels, reduce to the standard fractional operators.…”