Generalized fractional calculus with applications to the calculus of variations

Abstract

We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental and isoperimetric problems, natural boundary conditions, and Noether type theorems.

Fractional calculus of variations of several independent variables

“…Our goal is to develop a theory of the fractional calculus of variations by considering multidimensional fractional variational problems with Lagrangians depending Our results generalize the fractional calculus of variations for functionals involving multiple integrals studied in [3,26,31], as well as previous works about extremizers of single variable integral functionals with generalized fractional operators [2,23,24].…”

“…By the choice of an appropriate kernel, these operators can be reduced to the standard fractional integrals and derivatives. For more on the subject we refer the reader to [2,14,23,24].…”

“…Operators A, B and K reduce to the classical fractional integrals and derivatives for suitably chosen kernels and p-sets (see [23,24]). The notation was introduced in [2] and is now standard [16,25].…”

Reflection symmetric formulation of generalized fractional variational calculus

“…These early results demonstrate the main difficulties one meets c 2013 Diogenes Co., Sofia pp. 243-261 , DOI: 10.2478/s13540-013-0015-x when decribing physical phenomena via minimum action principle and including memory effects into a model (compare results and comments in [2]- [13], [18], [23]- [26]). The obtained Euler-Lagrange equations are non-local and mix derivatives determined by the left and right neighbourhood of the given time point.…”

Fractional variational approach with non-standard power-law degenerate Lagrangians and a generalized derivative operator

“…One possible extension of the calculus of variations is based on the fractional calculus. In fact, in recent years, there has been lot of works dedicated to the fractional actionlike variational approach with fractional derivatives and fractional integrals where different forms of the fractional Euler-Lagrange equations were obtained depending on the type of fractional functional or fractional Lagrangian systems used [1,3,26,27,31,39,40]. Most recent and broadest overviews of applications of the fractional calculus of variations are found in [34,36].…”