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

Definitions of fractional derivative of order $α$ ($0 < α\leq 1$) 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 $α= 1$). In order to satisfy an arbitrary initial condition the definitions of fractional derivative must necessarily involve a singular kernel.

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