A real regularised fractional derivative

A real regularised integral formulation of the frac- tional derivative is obtained from the generalised Grünwald– Letnikov derivative without using the Cauchy derivative. This new approach is based on the properties of the Mellin transform. The usual Riemann–Liouville and Caputo derivatives are expressed in a similar way emphasising their regu- larising capabilities. Some examples involving the Heaviside unit step function are presented in the last section of the paper.