Recently, some new operators having the keyword "fractional" are proposed to define the non-integer order derivative.
For integer order derivatives, it is well known that the locality and validity of Libeniz rule are the main properties of integer order derivatives, but what is the characteristic property of a fractional derivative? When we can call an operator is "fractional" and what features should an operator have to call it a "derivative" operator?
This is a not simple question. As it is normal, researchers have different opinions and you will not find a definitive answer. Anyway, some authors have started to debate and examine the question in a rigorous way. I would like to suggest the reading of these papers which are the state-of-the-art and the starting point for this discussion.
Ortigueira and Machado
Article What is a fractional derivative?
Tarasov
Article No Nonlocality. No Fractional Derivative
Ortigueira and Machado
Article A critical analysis of the Caputo-Fabrizio operator
Tarasov
Article Local Fractional Derivatives of Differentiable Functions are...
From the functional analytic point of view a fractional derivative is a fractional power of a differential operator. It is doubtful that there is any other characteristic property of a fractional derivative in my opinion.
Best regards,
A. M.
A linear thought often may not be a true or good idea or constructive. I think some new definitions should have been considered by more attention. Please see the following papers:
Article Response functions in linear viscoelastic constitutive equat...
Article On the existence of solutions for some infinite coefficient-...
Article Fractional derivatives with no-index law property: Applicati...
I recommend the history of science, how sqrt(-1)=i is accepted, how the Non-Euclidean geometry is accepted, and how the odd theory of relativity is confirmed. Still, it is begging of many new ideas.
Babak Shiri
The geometrical and physical meaning of ordinary derivative is simple and intuitive: For smooth function f which is differentiable at x, the local behavior of f around point x. The ordinary derivative gives the linear approximation of smooth function. Here we expect the fractional order derivative to have the similar geometrical meaning. We hope for non-differentiable functions, the fractional order derivative could give some kind approximation of its local behavior. A simple definition directly from geometrical meaning: We expect that the fractional derivative could give nonlinear (power law) approximation of the local behavior of non-differentiable functions. Fractional dynamics is the field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power law non-locality, power law long-term memory or fractal properties by using integrations and differentiation of an arbitrary real order.
One paper I encountered awhile back attempted to approach this from the point of view of outlining criteria of which may be used to classify so called fractional derivatives.
The paper was appropriately titled "What is a fractional derivative". They outline two sets of criteria referred to as wide sense and strict sense. Both of their criteria incorporate a need to satisfy a generalized Leibniz derivative rule, and they build upon much earlier work by Bertran Ross in 1975 (although he neglected to use the notion of a generalized Leibniz derivative rule).
A second, more recent paper titled "No non-locality. No Fractional Derivative" addresses the requirement that non-locality be incorporated within the fractional derivative.
To the best of my knowledge there is no strict consensus regarding the particular properties required for fractional derivatives. However, in my mind the two articles I have made reference to serve to provide a reasonable set of properties for one to expect fractional derivative operators to possess.
All the best.
Dear colleagues
Interesting question and I want to add a reply, about the Leibnitz Rule. You can build local derivatives that do not satisfy it! Conclusion: in my opinion there is no clear classification in this regard. I prefer a broader sentence: fractional derivatives are not derivatives, local generalized derivatives are not fractional.
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