An extension of Caputo fractional derivative operator and its applications

Abstract

In this paper, an extension of Caputo fractional derivative operator is introduced, and the extended fractional derivatives of some elementary functions are calculated. At the same time, extensions of some hypergeometric functions and their integral representations are presented by using the extended fractional derivative operator, linear and bilinear generating relations for extended hypergeometric functions are obtained, and Mellin transforms of some extended fractional derivatives are also determined.

Extended Riemann-Liouville type fractional derivative operator with applications

“…Also many extensions of fractional derivative operators have been developed and applied by many authors (see [2-6, 11, 12, 19-21] and [17,18]). These new extensions have proved to be very useful in various elds such as physics, engineering, statistics, actuarial sciences, economics, nance, survival analysis, life testing and telecommunications.…”

On some subclasses of hypergeometric functions with Djrbashian Cauchy type kernel

“…One can be convinced that the above defined functions become those of [11] for ω(t, p, 0) = e −p t(1−t) and Re p > 0, while for ω ≡ 1 these functions become the well-known Gauss hypergeometric function 2 F 1 and the Appell function F 2 , respectively. Definition 2.3.…”

Incomplete Caputo fractional derivative operators

“…Note that when m = 0, these functions reduce to the corresponding versions given by (14), (15), (18)- (21), respectively. On the other hand, in the case y → 1 -, the functions in (24), (26), (28), and (30) are reduced to their usual versions (similarly, as y → 0 + , the functions in (25), (27), (29), and (31) are reduced to their usual versions).…”

“…Çetinkaya [7] introduced the incomplete second Appell hypergeometric functions by means of the incomplete Pochhammer symbols and obtained some integral representations and transformation formulas for these functions. After these works, incomplete hypergeometric functions have become one of the hot topics of recent years [4,7,11,12,14,15,21,23,25,26,[29][30][31][32].…”

“…and they gave linear and bilinear generating relations for incomplete hypergeometric functions defined in (14) and (15 …”