What is really a fractional derivative? When can we affirm it is? Shouldn't we require that they recover the classic when the order becomes integer?
The word fractional appears in a lot of contexts. It became like a fashion. It is clear that Grunwald-Letnikov, Riemann-Louville, Caputo, or Riesz are fractional derivatives, but it is clear for me that the so called fractional Fourier transform is not really fractional. And like this one there are others. Personally, I believe that the most correct way of going into the fractional derivative is the Grunwald-Letnikov, because all the others can be deduced from it. Besides it is the most suitable for numerical computations.
The sixth FDA workshop, FDA'2013 (FDA : Fractional differentiation and
its Applications) will be held in Grenoble, France, February 4-6, 2013
in the SSSC joint conferences (Track D of this conference). All the
important dates of this workshop are in the attached file Papers
submission deadline June 15th, 2012) . As for all IFAC events, I remind
that all the papers will be peer-reviewed. The reviews will be done on a
6 pages draft (full paper, no abstract).
More information also available on the conference web site :
http://www.gipsa-lab.fr/SSSC2013/
Invited sessions within the technical scope of the conference are also
solicited. These proposals should contain six (6) full papers, or five
(5) including a survey paper limited to 12 pages
Thankyou so much Prof. for such a nice information about the FrFT and FOC. Waiting for such more information.
Dear Thierry
Thanks for your participation. My idea is to try to bring people to discuss the alternatives to Grunwald-Letnikov that I consider the most coherent. I considered the approach you suggest in one paper some time ago. There are three ways of doing it that lead to three different derivatives: one causal and two acausal (two-sided, equivalent to the Riesz potentials). The one you talked is the causal. The others are obtained by multiplication by |k|^a and |k|^a.sgn(k).
If you want papers on the subject I can send you [email protected]
Kolwankar K.M.; “Studies of Fractal Structures and Processes Using Methods of Fractional Calculus”;
http://arxiv.org/pdf/chao-dyn/9811008.pdf
Kolwankar K.M., and Gangal A.D.; "Local Fractional Derivatives and Fractal functions of several variables";
http://arxiv.org/pdf/physics/9801010.pdf
About fractional calculus I recommend the work of the Russian V. E. Tarasov, eg, Annals of Physics 323 (2008) 2756-2778
Fractional vector calculus and fractional Maxwell's equations
Vasily E. Tarasov *
Skobel'tsyn Institute of Nuclear Physics, Moscow State University, Leninskie gory, Moscow, Russia
The sixth FDA workshop, FDA'2013 (FDA : Fractional differentiation and
its Applications) will be held in Grenoble, France, February 4-6, 2013
in the SSSC joint conferences (Track D of this conference). All the
important dates of this workshop are in the attached file Papers
submission deadline June 15th, 2012) . As for all IFAC events, I remind
that all the papers will be peer-reviewed. The reviews will be done on a
6 pages draft (full paper, no abstract).
More information also available on the conference web site :
http://www.gipsa-lab.fr/SSSC2013/
Invited sessions within the technical scope of the conference are also
solicited. These proposals should contain six (6) full papers, or five
(5) including a survey paper limited to 12 pages
I agree with Prof. Ortiguera that the true fractional derivative is the Grunwald-Letnikov's because it is derived purely by derivative process unlike the Rieman-Liouvile that derives through integration process.
of course the fractional case is supposed to reduce to the classical case at integer value of the fractional order.
Thanks to all for these discussion.
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