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Remarks on fractional derivatives. (English) Zbl 1125.26009

The authors give a historical sketch of fractional calculus, which is readily available in details in the references given. On the other hand, they should also have mentioned the monograph “The fractional calculus. Theory and applications of differentiation and integration to arbitrary order” (1974; Zbl 0292.26011) by K. B. Oldham and J. Spanier, which happens to be the maiden text made available to researchers of this area of research. Moreover, this book contains an excellent chronological bibliography on fractional calculus by B. Ross (pp. 3–15).

The authors study in the present paper some properties of fractional derivatives, which is claimed to be interesting and new (not found elsewhere) by the authors. Grünwald-Letnikov fractional derivative, Riemann-Liouville fractional derivative and Caputo derivative are studied here. The Riemann-Liouville and the Caputo derivatives are compared and, further the sequential property of Caputo derivative is derived and simultaneously authors compare these two (mentioned above) derivatives with the classical derivative. The authors also give a sketch map, which illustrates consistency of fractional derivatives or integrals, which appears to be useful for some problems which require geometrical interpretation.

26A33Fractional derivatives and integrals (real functions)