On the generalized fractional derivatives and their Caputo modification

Abstract

In this manuscript, we define the generalized fractional derivative on AC n γ [a, b], the space of functions defined onWe present some of the properties of generalized fractional derivatives of these functions and then we define their Caputo version.

Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives

“…Remark 5.1 The oscillation of fractional differential equations in the frame of Katugampola-type fractional derivatives studied in [23][24][25] can be investigated in a similar way as we have done in this article for CFDs and their Caputo settings. The reader can verify sufficient conditions and the proofs by observing the kernel which is free from the starting point a.…”

“…We shall name this derivative the conformable fractional derivative (CFD). Although both the CFD with its Caputo setting and the Katugampola-type derivative studied in [23][24][25] coincide when a = 0, they are very different from each other. In fact, the kernel of CFDs depends on the end points a and b which causes many differences from the Katugampola-type one.…”

Analysis of some generalized ABC – Fractional logistic models

“…Despite this, these researchers have not stopped searching for more fractional operators, not only to enrich this calculus by discovering new kinds of fractional operators, but to understand better the complex systems they face in modeling. It can be realized that starting from the turn of this century researchers have proposed a variety of fractional operators [10,11,12,13,14,15,16] and added variety of fractional operators with different approaches to the field of fractional calculus.…”

Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives

“…The fractional calculus was bounded up with fractional integrals obtained by iterating an integral to get the nth order integral and then replacing n by any number, and then by using the classical method the corresponding derivatives were defined (see, for example, [1,[9][10][11][12][13][14]). However, for the sake of better description of real world phenomena, some scientists discovered new fractional operators with nonlocal and nonsingular kernels using the limiting process with the help of the Dirac delta function.…”