On the k-Riemann-Liouville fractional integral and applications

Abstract

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non - integer order. The subject is as old as differential calculus and goes back to times when G.W. Leibniz and I. Newton invented differential calculus. Fracti onal integrals and derivatives arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium. Very recently, Mubeen and Habi bullah have introduced the k - Riemann - Liouville fractional integral defined by using the - Gamma function, which is a generalization of the classical Gamma function. In this paper, we presents a new fractional integration is called k - Riemann - Liouville fractional integral, which generalizes the k - Riemann - Liouville fractional integral. Then, we prove the commutativity and the semi - group properties of the k - Riemann - Liouville frac tional integral and we give Chebyshev inequalities for k - Riemann - Liouville fractional integral. Later, using k - Riemann - Liouville fractional integral, we establish some new integral inequalities.