Could you explain me the use of fractional derivatives?

Solution 1:

It happens that I just see this old question. For such a long time, I suppose that you acquired as much useful information as expected.

A long time ago, I was interested on this subject. The book "The Fractional Calculus" from Keith B.Oldham, Jerome Spanier, Academic Press, New York, 1974 was of much greater use. Since that time, the number of publications is extensive.

I would mention this table, from https://fr.scribd.com/doc/14686539/The-Fractional-Derivation-La-derivation-fractionnaire , which shows a generalization of the basic electrotechnical components, thanks to the fractional differ-integration :

This is an interresting generalization especially in the field of impedance analysis. It makes understandable why the semi-circular complex impedance diagrams, with the center of circle depressed below the real axis, are so commonly observed in practice : https://fr.scribd.com/doc/71923015/The-Phasance-Concept