Fractional Line Integral

Manuel Ortigueira,Gabriel Bengochea

doi:10.3390/math9101150

Abstract

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

Highlights

It is no use to refer to the great evolution that made fractional calculus invade many scientific and technical areas [1,2,3,4]
This generalization was motivated by the results presented in [6], where classic theorems of vectorial calculus were introduced, but for integrations over rectangular lines
By using a standard procedure consisting of approximating a curve by a sequence of piecewise linear paths, we introduced the integration over any simple rectifiable line

Summary

Introduction

It is no use to refer to the great evolution that made fractional calculus invade many scientific and technical areas [1,2,3,4]. We tried to fill in another gap, by introducing a definition of the fractional line integral. This generalization was motivated by the results presented in [6], where classic theorems of vectorial calculus were introduced, but for integrations over rectangular lines. We opted for the directional derivatives resulting from the generalization of the Grünwald–Letnikov (GL) and Liouville (L) directional derivatives These were introduced and their main properties listed.

Functional Framework

Suitable Fractional Derivatives

Order-One Definite Integral

Fractional Definite Integral

The Grünwald–Letnikov and Liouville Directional Derivatives

On the Fractional Line Integrals

Conclusions