The next meeting of the Joint Seminar of Analysis, Geometry and Topology Department will be held on March 10, 2015 at 14:00 in Room 478 of IMI.

A talk on:

Generalized fractional derivatives of Riemann-Liouville  and Caputo type

will be delivered by Prof. Virginia Kiryakova.

Everybody is invited.

Abstract:

In Fractional Calculus (FC), as in the (classical) Calculus, the notions of derivatives and integrals (of first, second, etc. or arbitrary, incl. non-integer order) are co-related. One of the most frequent approach in FC is to define first the Riemann-Liouville (R-L) integral of fractional order, and then by means of suitable integer-order differentiation operation applied over it (or under its sign) a fractional derivative is defined – in the R-L sense (or in Caputo sense). The first mentioned (R-L type) is closer to the theoretical entertainments in analysis, but has some shortages – from the point of view of interpretation of the initial conditions for Cauchy problems (stated also by means of fractional order derivatives/ integrals), and also for the analysts’ confusion that such a derivative of a constant is not zero in general. The Caputo (C-) derivative, arising in geophysical studies, helps to overcome these problems and to describe models of applied problems with physically consistent initial conditions. Let us mention however that recently some authors dispute the advantages of the C-derivative against the R-L one, with examples from control theory.

The operators of the generalized fractional calculus (integrals and derivatives) represent commuting m-tuple (m = 1, 2, 3,...) compositions of operators of the classical FC with power weights (the so-called Erdelyi-Kober operators), represented in compact and explicit form by means of integral, integro-differential (R-L type) or differential-integral (C- type) operators, where the kernels are special functions of most general hypergeometric kind.

In this survey we present the genesis of the definition of the generalized fractional derivatives (of fractional multi-order) of R-L type, introduce the new ones of Caputo type, and analyze their properties and cases of coincidence of the definitions (for example for the hyper-Bessel differential operators or order m = multi-order (1, 1, …, 1), and for the Gelfond-Leontiev generalized differentiation operators). We consider some more particular examples of the derivatives of both types and of Cauchy problems for fractional order differential equations with R-L or C-derivatives and initial conditions of the corresponding type. Note that quite natural, the solutions of such problems are related to the Mittag-Leffler function or its multi-index analogues.