Jarad et al. Advances in Diﬀerence Equations (2017) 2017:247
R E S E A R C H Open Access
On a new class of fractional operators
, Thabet Abdeljawad
and Dumitru Baleanu
Department of Mathematics,
Çankaya University, Ankara, 06790,
Institute of Space Sciences,
Full list of author information is
available at the end of the article
This manuscript is based on the standard fractional calculus iteration procedure on
conformable derivatives. We introduce new fractional integration and diﬀerentiation
operators. We deﬁne spaces and present some theorems related to these operators.
Keywords: conformable derivatives; fractional conformable integrals; fractional
In the area of fractional calculus and its applications in many branches of science and engi-
neering, several fractional derivatives were mainly utilized. The most common used were
Caputo and Riemann-Liouville derivatives, which were successfully utilized in modeling
complex dynamics appearing in physics, biology, engineering and many other ﬁelds [–].
As is well known, systems possessing a memory eﬀect often appear in real world phenom-
ena. However, for each type of data we always ask what is the optimal corresponding non-
local model to be applied. Moreover, many authors studied new fractional operators with
local, nonlocal, singular and non-singular kernels (see [–] and the references therein).
The standard fractional calculus may not provide us the required kernel in order to ex-
tract important information from these types of systems. At this stage, we ask the follow-
ing question. Can we generalize the standard fractional Riemann-Liouville integrals in a
way such that we obtain uniﬁcation to Riemann-Liouville, Hadamard and other fractional
derivatives [, ]. The core of this procedure is to decide which diﬀerentiation operator
should be used as a starting point for the iteration procedure. For the standard fractional
calculus, we iterate the usual integral of a function and using the Cauchy formula we ob-
tain the integral of higher integer orders and then replace this integer by any complex
number. In , it was suggested that the conformable integral should be fractionalized
properly. We recall that an integral type like the one from  has appeared already in .
The integral mentioned below in () appears in mathematical economics, namely they are
used for describing discounting economical dynamics . Also,this integral appears in
describing the non-linear dissipative systems .
At this point we should say that the left and right conformable derivatives deﬁned in
, respectively, as
f (x)=(x – a)
f (x)=(b – x)
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