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

Abstract

Definitions of fractional derivative of order $\alpha$ ($0<\alpha \le 1$) 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.

References and extra comments added. 9 pages

References in corpus (3)

• The Prabhakar or three parameter Mittag--Leffler function: theory and application
• Some aspects of fractional diffusion equations of single and distributed order
• General fractional calculus and Prabhakar's theory