On the mistake in defining fractional derivative using a non-singular kernel

Abstract

Definitions of fractional derivative of order $\alpha$ ($0 < \alpha \leq 1$) using non-singular kernels have been recently proposed. In this note we show that these definitions cannot be useful in modelling problems with a initial value condition (like, for example, the fractional diffusion equation) because the solutions obtained for these equations do not satisfy the initial condition (except for the integer case $\alpha = 1$). In order to satisfy an arbitrary initial condition the definitions of fractional derivative must necessarily involve a singular kernel.