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

A. Torres-Hernandez, F. Brambila-Paz

TL;DR

This work proposes a set-based framework for fractional calculus, unifying a wide range of fractional operators into operator-sets and using them to define a fractional fixed-point methodology. It proves that each convergent fractional fixed-point method yields an uncountable family of convergent methods and provides a procedure to numerically estimate the mean order of convergence via critical-point problems, including a hybrid fractional iterative approach. The framework extends to complex spaces and encompasses fractional quasi-Newton, fractional pseudo-Newton, and fractional zeta variants, linking the fixed-point theory to optimization of scalar functions. The results suggest a broad, set-based theory of differential equations, potentially renaming fractional calculus as fractional calculus of sets.

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 $Ω$ 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.

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

TL;DR

This work proposes a set-based framework for fractional calculus, unifying a wide range of fractional operators into operator-sets and using them to define a fractional fixed-point methodology. It proves that each convergent fractional fixed-point method yields an uncountable family of convergent methods and provides a procedure to numerically estimate the mean order of convergence via critical-point problems, including a hybrid fractional iterative approach. The framework extends to complex spaces and encompasses fractional quasi-Newton, fractional pseudo-Newton, and fractional zeta variants, linking the fixed-point theory to optimization of scalar functions. The results suggest a broad, set-based theory of differential equations, potentially renaming fractional calculus as fractional calculus of sets.

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 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.

Paper Structure

This paper contains 7 sections, 4 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fixed Point Method
  3. Riemann-Liouville Fractional Operators
  4. Fractional Fixed Point Method
  5. Approximation to the Critical Points of a Function
  6. Examples
  7. Conclusions

Key Result

Proposition 2.2

Let $\Phi: \mathbb{R} ^n \to \mathbb{R} ^n$ be an iteration function that defines a sequence $\left\{x_i\right\} _{i\geq 1}$ such that $x_i\to \xi\in \mathbb{R} ^n$. So, if $\Phi$ has an order of convergence of order (at least) $p$ in $B(\xi;\delta)$, there exists a non-negative constant $K=K(C)$, s where $\left\lVert x_{-1} \right\rVert :=0$.

Figures (1)

  • Figure 1: Illustration of some lines generated by the fractional Newton-Raphson method for the same initial condition $x_0$ but with different orders $\alpha$ of the fractional operator implemented torres2021fracnewrap.

Theorems & Definitions (11)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • Corollary 4.2
  • Example 1
  • Example 2
  • ...and 1 more