A Study on Fractional Taylor Series and Variations

Fractional mechanics has been recently one of the most efficient branches of mechanics, interpreting successfully various experiments. Two of the most powerful tools for solving mechanics problem, i.e Taylor’s series and variational approaches are discussed in the context of fractional analysis.Λ-fractional derivative is introduced and fractional Taylor’s series is established, along with the fractional calculus of variations. In addition to the aforementioned derivative the according fractional Λ -space is defined in which the derivative satisfies all the conditions of a derivative required by differential topology. In fact the fractional derivatives in the initial space correspond to the Λ-derivatives in the Λ-space, behaving like the conventional ones in that space. Since geometry is feasible only in the Λ -fractional space, the analysis is performed in that space. Then the results are also derived in the initial space. The fractional beam bending problem is addressed as an application of fractional calculus of variations. Further application involving combination of the fractional variational approach and fractional Taylor’s series is presented, analyzing the fractional stability and post-stability problem of a rod, under axial compression