A unified approach to fractional derivatives

A new fractional derivative of the Grüwald–Letnikov type is proposed and some properties are studied. The new definition incorporates both the forward and backward Grüwald–Letnikov and the two-sided fractional derivatives. Several properties of such generalized operator are presented, and some particular cases deduced.

► The new derivative of the incremental ratio type incorporates 4 existing fractional derivatives.
► It leads to a general integral representation having the Liouville and Riesz integrals as special cases.
► It allows us writing the partial differential equations with a unique derivative formulation.
► It exists for a wide class of functions.
► It is suitable for numerical implementations.