Analytical schemes for a new class of fractional differential equations

Fractional differential equations (FDEs) considered so far contain mostly left (or forward) fractional derivatives. In this paper, we present analytical solutions for a class of FDEs which contain both the left and the right (or the forward and the backward) fractional derivatives. The methods presented use properties of fractional integral operators (which, in many cases, lead to Volterra-type integral equations), an operational approach and a successive approximation method to obtain the solutions. The methods are demonstrated using some examples. The FDEs considered may come from fractional variational calculus (FVC) or from other physical principles. In the case of fractional variational problems (FVPs), the transversality conditions are used to identify appropriate boundary conditions and to solve the problems. It is hoped that this study will lead to further investigations in the field and more elegant solutions would be found.