Fractional calculus: What is it?

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. The subject is as old as the differential calculus, and goes back to times when Leibniz and Newton invented differential calculus. The most common notations for β-th order derivative of a function  y(t)  defined in  (a,b)  are  y^(β)(t)  and   _aD_t^βy(t). Negative values of β correspond to fractional integrals.

In a letter to L`Hôpital in 1695 Leibniz raised the following question: "Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?" L`Hôpital was somewhat curious about that question and replied by another question to Leibniz: "What if the order will be 1/2?" Leibniz in a letter dated September 30, 1695 — the exact birthday of the fractional calculus! — replied: "It will lead to a paradox, from which one day useful consequences will be drawn." The question raised by Leibniz for a fractional derivative was an ongoing topic for more than 300 years. Many known mathematicians contributed to this theory over the years, among them Liouville, Riemann, Weyl, Fourier, Abel, Lacroix, Leibniz, Grunwald and Letnikov. For a historical survey the reader may consult the books by K.B. Oldham and J. Spanier or K.S. Miller and B. Ross.

Further reading

• Podlubny I.: Derivácie neceločíselného rádu: história, teória, aplikácie. Plenárna prednáška na konferencii Slovenskej matematickej spoločnosti, Jasná, 23. novembra 2003. (Fractional-order derivatives: History, theory, and applications. Plenary lecture at the Conference of the Slovak Mathematical Society, Jasná, November 23, 2002, in Slovak. — [90 slides, PDF (7.51 MB)]
• Fractional Calculus Applications in Automatic Control and Robotics. Tutorial Workshop at the 41st IEEE Conference on Decision and Control.
• See also a list of books related to fractional calculus