Azam et al. Advances in Diﬀerence Equations (2017) 2017:249
R E S E A R C H Open Access
Study of generalized type K-fractional
University of Management and
Technology, Lahore, Pakistan
Full list of author information is
available at the end of the article
In this paper, the generalized type k-fractional derivatives are introduced and their
semi-group, commutative and inverse properties are presented. These derivatives can
be reduced to other fractional derivatives by substituting the values of the
parameters involved. The Mellin transform of generalized Caputo type k-fractional
derivative is also found.
MSC: 42A38; 26A33
Keywords: generalized k-fractional derivative; generalized Caputo type k-fractional
derivative; Mellin transform
Fractional calculus is the study of theory and applications of derivatives and integrals of
non-integer order. It is a generalized form of calculus, so it retains many properties of
calculus. It is worth mentioning that, in recent times, theory of fractional calculus has
developed quickly and played many important roles in science and engineering, serving as
a powerful and very eﬀective tool for many mathematical problems. It has been extensively
investigated in the last two decades.
Fractional derivatives are of vital importance in fractional calculus. These fractional
derivatives are used in mathematical physics, astrophysics, control theory, electric con-
ductance of biological systems, statistical mechanics, ﬁnance, biophysics, electrochem-
istry, computed tomography, geological surveying, thermodynamics, hydrology and engi-
neering; moreover, they are also drawn on for the mathematical modelling of viscoelastic
Fractional derivatives have also been employed recently in signal and image possessing.
They also have a key role in electric conductance of biological systems and fractional or-
der models of neurons. The application of fractional order derivatives to the modelling of
diﬀusion in a speciﬁc type of porous medium is in practice as well.
The objective and motivation of this work is to develop new generalized type k-
fractional derivatives, which are the generalized form of the existing fractional derivatives,
as well as to highlight the importance of their applications in diverse research areas. The
generalized k-fractional and generalized Caputo type k-fractional derivatives are the gen-
eralized forms of some existing fractional derivatives.
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