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.

arXiv commentReferences and extra comments added. 9 pages

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Cited by in corpus (1)

• The Exact Spectral Derivative Discretization Finite Difference (ESDDFD) Method for Wave Models: A Universal Wave View Through Natural Fractional / Fractal Derivative Representations (or View Lens Shops for The Exponential Wave Universe)