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 formula_1 be the space of those Lebesgue measurable functions formula_2 on formula_3 for which formula_4, where the norm is defined by formula_5 for formula_6 and for the case formula_7 formula_8 Katugampola fractional integral. It is defined via the following integrals for formula_9 and formula_10 This integral is called the "left-sided" fractional integral. Similarly, the "right-sided" fractional integral is defined by, for formula_11 and formula_12. These are the fractional generalizations of the formula_13-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 formula_17 and formula_18 The generalized fractional derivatives, corresponding to the generalized fractional  integrals () and () are defined, respectively, for formula_19, 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, formula_22, 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 formula_23 and formula_24. The C-K fractional derivative of order formula_25 of the function formula_26 with respect to parameter formula_24 can  be expressed as It satisfies the following result. Assume that formula_29, 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 formula_31 and type formula_32. The fractional derivative (left-sided/right-sided), with respect to formula_33, with formula_34, is defined by where formula_36, for functions formula_37 in which the expression on the right hand side exists, where formula_38 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 formula_39 and formula_18 Then, formula_41 for formula_42, if formula_43 exists for formula_44. Hermite-Hadamard type inequalities. Katugampola operators satisfy the following Hermite-Hadamard type inequalities: Theorem. Let formula_45 and formula_46. If formula_2 is a convex function on formula_3, then formula_49 where formula_50. When formula_51, in the above result, the following Hadamard type inequality holds: Corollary. Let formula_45. If formula_2 is a convex function on formula_3, then formula_55 where formula_56 and formula_57 are left- and right-sided Hadamard fractional integrals. Recent Development . These operators have been mentioned in the following works: Notes. The CRONE (R) Toolbox, a Matlab and Simulink Toolbox dedicated to fractional calculus, can be downloaded at http://cronetoolbox.ims-bordeaux.fr