In 2016 a new definition of a fractional derivative was announced in this paper, which has since had more than 100 citations. This derivative, the Atangana-Baleanu derivative, is the main recent development in the field of fractional calculus.

How important is it? Does it merit being given the same weight as the Riemann-Liouville and Caputo fractional derivatives, and has it the potential to raise the importance of fractional calculus itself in the broader field of analysis?

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    $\begingroup$ I've deleted the other question (erasing the contents does not count as a deletion; you have to hit the delete button). For future reference: better practice is to clarify the old question rather than post another version of it. $\endgroup$ – Todd Trimble Oct 21 '17 at 15:14
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    $\begingroup$ @AlexM - This q cannot be answered, let alone tell an interested person all they wish to know, with reference only to what universities the authors work at and what text or document processing program they use. If you wish to put those considerations aside and look dispassionately and disinterestedly (but not uninterestedly!) at the work asked about, I'd be interested to hear your thoughts and conclusions. If this new derivative is probably going nowhere because of Hörmander's work on pseudodifferential operators, an answer in which you try to justify (and nuance) that idea would be welcome. $\endgroup$ – willhart Oct 21 '17 at 18:15
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    $\begingroup$ This question increasingly seems like a fishing expedition with the OP wanting people to validate this particular field of study. Negative opinions seem to be received reluctantly $\endgroup$ – Yemon Choi Oct 21 '17 at 21:21
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    $\begingroup$ @YemonChoi - You said in its other form it was a magnet for debates about people's merits; now you say I'm seeking validation and I (because I assume it's me you're talking about) receive negative opinions reluctantly. You're so wrong. I'm seeking answers to the question that are backed up with statements about the particular work mentioned, which as I said is the main recent development in the field. I don't mind whether they say the work is rubbish and leads nowhere or it is brilliant and has enormous potential, just so long as the conclusions are supported by comments on what it says. $\endgroup$ – willhart Oct 21 '17 at 21:47
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    $\begingroup$ Surely the main current development in FC is an excellent testing ground for the idea that FC cannot yield major insights or "breakthroughs" that have not already in effect been discovered by researchers into pseudodifferential operators or at least that do not appear as relatively uninteresting and not especially insightful when looked at from an angle informed by an understanding of pseudodifferential operators or of other advanced already known work in analysis? This is what the question is about, and my mind is completely open as to what the right answer is :-) $\endgroup$ – willhart Oct 22 '17 at 9:56

While the operators in question may be of some use in applied mathematics, in my opinion they bring little new to the theory of fractional calculus.

Both variants discussed in the linked paper are of the form $$ D^\alpha f(t) = a f(t) + b f'(t) + \int_{-\infty}^t (f(t) - f(s)) k(t - s) ds $$ for some kernel $k$ and some constants $a$ and $b$, with a proper extension of $f$ from the half-line $(b,\infty)$ to all of $\mathbb{R}$ (the "boundary term"). Operators of this form are very well-studied; for example, in probability, these are generators of Lévy processes.

The usual Riemann–Liouville (or Caputo) derivative $D_R^\alpha$ has kernel $k$ with Fourier transform $\xi^\alpha$. Probabilistically, $D_R^\alpha$ corresponds to the $\alpha$-stable subordinator (a subordinator is an increasing Lévy process).

The kernel $k$ used in the paper has Laplace transform equal to $$ \int_0^\infty e^{-\xi t} k(t) dt = \frac{\xi^\alpha}{\xi^\alpha + \tfrac{\alpha}{1 - \alpha}} ;$$ see formula (9) there. The corresponding operator $D^\alpha_{AB}$ can be written as a composition of the usual derivative $D_R$ and the $\lambda$-resolvent operator $(\lambda + D_R)^{-1}$. This is a rather standard object in the theory of strongly continuous semigroups of operators.

The operator $D^\alpha_{AB}$ indeed resembles fractional differentiation in large scales. However, at a small scale, this is really loosely related to fractional differentiation. In particular, (essentially) no regularity of $f$ is needed in order to evaluate $D^\alpha_{AB} f$. This is not a property one would expect from a "fractional derivative".

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