A new definition of fractional derivative

We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for 0 <= alpha < 1 coincides with the classical definitions on polynomials (up to a constant). Further, if alpha = 1, the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations. (C) 2014 Elsevier B.V. All rights reserved.