Fractional Derivative as Fractional Power of Derivative

TL;DR

This paper introduces new definitions of fractional derivatives as fractional powers of derivative operators using Fourier and Taylor series, and explores their implications for stability analysis.

Contribution

It proposes novel definitions of fractional derivatives based on fractional powers of operators, utilizing Fourier series, Fourier integrals, and Weyl quantization.

Findings

Defined fractional derivatives via Fourier and Taylor series

Extended stability concepts to fractional derivatives

Provided operator-based fractional derivative formulations

Abstract

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.