Further properties of Osler's generalized fractional integrals and derivatives with respect to another function

Ricardo Almeida

doi:10.1216/rmj-2019-49-8-2459

Abstract

In this paper we discuss fractional integrals and fractional derivatives of a function with respect to another function. We present some fundamental properties for both types of fractional operators, such as Taylor's theorem, Leibniz and semigroup rules. We also provide a numerical tool to deal with these operators, by approximating them with a sum involving integer-order derivatives.

Highlights

Fractional calculus is an important research field, in pure mathematics, but in applied mathematics, physics, biology, engineering, economics, etc., as well
The subject is as old as calculus itself, and goes back to Leibniz and L’Hopital, when the meaning of the derivative of order 1/2 was discussed
Fractional operators do not share the same properties and because of this we find a large number of works for similar problems

Summary

Introduction

Fractional calculus is an important research field, in pure mathematics, but in applied mathematics, physics, biology, engineering, economics, etc., as well. [16] The left and right Riemann–Liouville fractional integrals of f of order α > 0, with respect to function ψ, are given by [16] The left and right Riemann–Liouville fractional derivatives of f of order α > 0, with respect to function ψ, are given by If f is a continuous function on [a, b], Iaα+,ψf (x) and Ibα−,ψf (x) are well defined for every x ∈ [a, b].

Results

Conclusion