Fractional Calculus - A Commutative Method on Real Analytic Functions

TL;DR

This paper introduces a new commutative operator framework for fractional calculus on real analytic functions, enabling more consistent and algebraic manipulation of fractional derivatives.

Contribution

It constructs a space and operators where fractional derivatives commute, providing a novel algebraic approach to fractional calculus on analytic functions.

Findings

Operators commute within the constructed space.

Embedding of analytic functions preserves fractional differentiation.

Simplifies fractional calculus operations through algebraic structures.

Abstract

The traditional first approach to fractional calculus is via the Riemann-Liouville differintegral $_{a}D_{x}^{k}$. The intent of this paper will be to create a space $K$, pair of maps $g: C^{\omega}(\mathbb{R}) \to K$ and $g': K \to C^{\omega}(\mathbb{R}$), and operator $D^{k}: K \to K$ such that the operator $D^{k}$ commutes with itself, the map $g$ embeds $C^{\omega}(\mathbb{R}$) isomorphically into $K$, and the following diagram commutes; \xymatrix{C^{\omega}(\mathbb{R}) \ar[d]_{_{a}D_{x}^{k}} \ar[r]^{g} & K \ar[d]^{D^{k}} C^{\omega}(\mathbb{R}) & K \ar[l]^{g'}} \qquad This implies the following diagram commutes, for analytic $f$ such that $_{a}D_{x}^{j}f$ = 0 (i.e, if $f = \sum_{i \in I}b_{i}$(x-$a)^{i}$, where {$b_{i}} \subset \mathbb{R}$, and $I \subseteq {j-1, ..., j-\lfloor j \rfloor$}); \xymatrix{f \ar@/_3pc/[dd]_{_{a}D_{x}^{j+k}} \ar[d]^{_{a}D_{x}^{j}} \ar[r]^{g} & g(f) \ar[d]^{D^{j}} 0 & \ar[l]^{g'} D^{j}g(f) \ar[d]^{D^{k}} _{a}D_{x}^{j+k}f &\ar[l]^{g'} D^{k}D^{j}g(f)}