Katugampola fractional operators

In mathematics, Katugampola fractional operators are integral operators that generalize the Riemann–Liouville and the Hadamard fractional operators into a unique form. The Katugampola fractional integral generalizes both the Riemann–Liouville fractional integral and the Hadamard fractional integral into a single form and It is also closely related to the Erdelyi–Kober operator that generalizes the Riemann–Liouville fractional integral. Katugampola fractional derivative has been defined using the Katugampola fractional integral and as with any other fractional differential operator, it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators.

Definitions

These operators have been defined on the following extended-Lebesgue space..
Let be the space of those Lebesgue measurable functions on for which, where the norm is defined by
for and for the case

Katugampola fractional integral

It is defined via the following integrals

for and This integral is called the left-sided fractional integral. Similarly, the right-sided fractional integral is defined by,

for and.
These are the fractional generalizations of the -fold left- and right-integrals of the form
and
respectively. Even though the integral operators in question are close resemblance of the famous Erdélyi–Kober operator, it is not possible to obtain the Hadamard fractional integrals as a direct consequence of the Erdélyi–Kober operators. Also, there is a corresponding fractional derivative, which generalizes the Riemann–Liouville and the Hadamard fractional derivatives. As with the case of fractional integrals, the same is not true for the Erdélyi–Kober operator.

Katugampola fractional derivative

As with the case of other fractional derivatives, it is defined via the Katugampola fractional integral.
Let and The generalized fractional derivatives, corresponding to the generalized fractional integrals and are defined, respectively, for, by
and
respectively, if the integrals exist.
These operators generalize the Riemann–Liouville and Hadamard fractional derivatives into a single form, while the Erdelyi–Kober fractional is a generalization of the Riemann–Liouville fractional derivative. When,, the fractional derivatives are referred to as Weyl-type derivatives.

Caputo–Katugampola fractional derivative

There is a Caputo-type modification of the Katugampola derivative that is now known as the Caputo–Katugampola fractional derivative.
Let and. The C-K fractional derivative of order of the function with respect to parameter can be expressed as
It satisfies the following result. Assume that, then the C-K derivative has the following equivalent form

Hilfer–Katugampola fractional derivative

Another recent generalization is the Hilfer-Katugampola fractional derivative. Let order and type. The fractional derivative,
with respect to, with, is defined by
where, for functions in which the expression on the right hand side
exists, where is the generalized fractional integral
given in.

Mellin transform

As in the case of Laplace transforms, Mellin transforms will be used specially when solving differential equations. The Mellin transforms of the left-sided and right-sided versions of Katugampola Integral operators are given by

Theorem

Let and Then,
for, if exists for.

Hermite-Hadamard type inequalities

Katugampola operators satisfy the following Hermite-Hadamard type inequalities:

Theorem

Let and. If is a convex function on, then
where.
When, in the above result, the following Hadamard type inequality holds:

Corollary

Let. If is a convex function on, then
where and are left- and right-sided Hadamard fractional integrals.