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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 genaralizes 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.
Contents
Definitions
These operators have been defined on the following extended-Lebesgue space.
Let
for
Katugampola fractional integral
It is defined via the following integrals
for
for
These are the fractional generalizations of the
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
respectively, if the integrals exist.
It is interesting to note that 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,
Caputo–Katugampola fractional derivative
There is a Caputo-type modification of the Katugampola derivative that is now known as the Caputo–Katugampola fractional derivative.
These operators have been mentioned in the following works:
- Fractional Calculus. An Introduction for Physicists, by Richard Herrmann
- Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics, Tatiana Odzijewicz, Agnieszka B. Malinowska and Delfim F. M. Torres, Abstract and Applied Analysis, Vol 2012 (2012), Article ID 871912, 24 pages
- Introduction to the Fractional Calculus of Variations, Agnieszka B Malinowska and Delfim F. M. Torres, Imperial College Press, 2015
- Advanced Methods in the Fractional Calculus of Variations, Malinowska, Agnieszka B., Odzijewicz, Tatiana, Torres, Delfim F.M., Springer, 2015
- Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Shakoor Pooseh, Ricardo Almeida, and Delfim F. M. Torres, Numerical Functional Analysis and Optimization, Vol 33, Issue 3, 2012, pp 301–319.
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
for
Hermite-Hadamard type inequalities
Katugampola operators satisfy the following Hermite-Hadamard type inequalities:
Theorem
Let
where
When
Corollary
Let
where