G-Meijer functions series as solutions for certain fractional variational problem on a finite time interval

Abstract

Simple fractional variational problem on a finite time interval is studied. This equation, obtained by minimum action principle, contains left- and right-sided fractional derivatives. When fractional order α ∈ 2 (1, 2) we arrive at variational version of fractional oscillator, which yields in the limit α → 1 + the classical harmonic oscillator. Mellin transform is applied and general continuous solution inform of G-Meijer functions series is derived. An approximate solution for fractional case and small values of λ - square of frequency is explicitly calculated. For integer order a = I classical results are recovered.