Sets of fractional operators and numerical estimation of the order of convergence of a family of fractional fixed point methods

TL;DR

This paper introduces a set-based classification of fractional operators, generalizes classical calculus objects, and proposes a numerical method to estimate the convergence order of fractional fixed point methods.

Contribution

It presents a novel fractional calculus framework using sets, generalizes key calculus concepts, and develops a numerical approach to estimate convergence order in fractional fixed point methods.

Findings

A simplified, compact fractional calculus framework using sets.

Generalization of classical calculus objects to fractional context.

A method to numerically estimate the convergence order of fractional fixed point methods.

Abstract

Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a simplified and compact way to work the fractional calculus through the classification of fractional operators using sets. This new way of working with fractional operators, which may be called as fractional calculus of sets, allows to generalize objects of the conventional calculus such as tensor operators, the diffusion equation, the heat equation, the Taylor series of a vector-valued function, and the fixed point method in several variables which allows to generate the method known as the fractional fixed point method. It is also shown that each fractional fixed point method that generates a convergent sequence has the ability to generate an uncountable family of fractional fixed point methods that generate convergent sequences. So, it is shown one way to estimate numerically the mean order of convergence of any fractional fixed point method in a region $\Omega$ through the problem of determining the critical points of a scalar function, and it is shown how to construct a hybrid fractional iterative method to determine the critical points of a scalar function.