Riemann-Liouville and Caputo type multiple Erdelyi-Kober operators

In this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erd,lyi-Kober operators. Then the multiple Erd,lyi-Kober fractional integrals and derivatives of R-L type of multi-order (delta (1),aEuro broken vertical bar,delta (m) ) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erd,lyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.