Without any doubt: Nishimoto made the major contributions to FC.

See my contributions too and atangana baleanu fractional derivative. Seriously many have done as Dr. Ndolane SENE and Atangana abdon. As Machado, Manual,....the list is long. Your question is not adequate. Please remove it.

Best personnal regards.

Without doubt Prof. Mokhtar Kirane is one of the most influential researchers on the topic

Thanks professor Ortigueira

Fethi belgacem do think in fractionnal too and antagana.

I forgot G. O. Okikiolu. In India there were several.

About Atangana, I read yesterday a very curious paper "Derivative with two fractional orders: A new avenue of investigation toward revolution in fractional calculus," It is an avenue with too many "holes". Formulae (3) and (4) are completely unacceptable since (x-t) is negative and its noninteger power is complex.

N. Sene wrote another interesting paper "Analytical solutions of electrical circuits considering certain generalized fractional derivatives" that would be used by Electrical Engineers as a joke: he made the miracle of creating a RLC circuit that does not oscillate, not even in a transient regime.

the problem with fractional derivative is the numerical solution of the equations. we use fourier transform or other operator bit the domain computation is at most square

Dear Mohamed

You are quite right, but, in many applications, we need not to solve the equations. Frequently, we are only interested in getting information on the behaviour of the systems, not solving an equation and this can be obtained in frequency domain. Anyway, the Signals and Systems approaches allow us to obtain many interesting results. The continuous to discrete time conversion allow also obtain good results, using for example the FFT.

I think it is worthwhile to mention the Indian mathematician Tilak Raj Pabhakar who proposed and studied a generalization to three parameters of the Mittag-Leffler function, see [T.R.Prabhakar, A singular integral equation with a generalized Mittag–Leffler function in the kernel, Yokohama Math. J., 19(1):7–15, 1971]. Although this work was not specifically devoted to fractional calculus, the now so-called Prabhakar function allows to solve in the time-domain fractional differential equations involved by the Havriliak-Negami model, a dielectric model which needs a special fractional derivative with two fractional parameters.

Thanks professor Garrappa