Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative

Abstract

The Caputo fractional derivative is one of the most used definitions of a fractional derivative along with the Riemann-Liouville and the Grünwald- Letnikov ones. Whereas the Riemann-Liouville definition of a fractional derivative is usually employed in mathematical texts and not so frequently in applications, and the Grünwald-Letnikov definition – for numerical approximation of both Caputo and Riemann-Liouville fractional derivatives, the Caputo approach appears often while modeling applied problems by means of fractional derivatives and fractional order differential equations. In the mathematical texts and applications, the so called Erdélyi-Kober (E-K) fractional derivative, as a generalization of the Riemann-Liouville fractional derivative, is often used, too. In this paper, we investigate some properties of the Caputo-type modification of the Erdélyi-Kober fractional derivative. The relation between the Caputo-type modification of the E-K fractional derivative and the classical E-K fractional derivative is the same as the relation between the Caputo fractional derivative and the Riemann-Liouville fractional derivative, i.e. the operations of integration and differentiation are interchanged in the corresponding definitions. Here, some new properties of the classical Erdélyi-Kober fractional derivative and the respective ones of its Caputo-type modification are presented together.