The General Fractional Integrals and Derivatives on a Finite Interval

Abstract

The general fractional integrals and derivatives considered so far in the Fractional Calculus literature have been defined for the functions on the real positive semi-axis. The main contribution of this paper is in introducing the general fractional integrals and derivatives of the functions on a finite interval. As in the case of the Riemann–Liouville fractional integrals and derivatives on a finite interval, we define both the left- and the right-sided operators and investigate their interconnections. The main results presented in the paper are the 1st and the 2nd fundamental theorems of Fractional Calculus formulated for the general fractional integrals and derivatives of the functions on a finite interval as well as the formulas for integration by parts that involve the general fractional integrals and derivatives.

Keywords

• Sonin kernels
• Sonin condition
• general fractional integral
• general fractional derivative
• fundamental theorems of fractional calculus