Fractional integration and differentiation

TL;DR

This paper presents a novel method for calculating fractional integrals and derivatives using an equation derived from infinite applied integration by parts, offering a series representation beneficial for smooth functions and approximations.

Contribution

Introduces a new series-based method for fractional calculus derived from infinite applied integration by parts, applicable to smooth functions.

Findings

Provides a series representation for fractional integrals and derivatives.

Applicable to a special class of functions with potential for approximation.

Reduces the value of sequential elements for smoother function analysis.

Abstract

In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of functions and provides a series representation of integration. This representation will be useful for working with smooth functions and for approximation due to the potential reduction of the value of the sequential elements.