On Fractals and Fractional Calculus Motivated by Complex Systems

TL;DR

This paper explores the application of fractional calculus to model complex systems with fractal structures, proposing new fractional equations and a generalized Weibull distribution to better describe their statistical properties.

Contribution

It introduces fractional generalizations of Schrödinger and Klein-Gordon equations inspired by fractal space-time theory and proposes a generalized Weibull distribution for heavy-tailed statistics in complex systems.

Findings

Fractional calculus better models fractal systems.

New fractional equations extend quantum mechanics.

Generalized Weibull distribution captures heavy-tailed behavior.

Abstract

Complex systems (CS) are ubiquitous in nature. It is argued that fractional order (FO) calculus is more suitable to describe fractal systems. Motivated by the fractal space time theory some fractional generalizations of Scrodinger and Klein-Gordon equations are given. In many CS systems statistics is described by heavy tailed distributions e.g. fractal and Levy-Weibull ones. Here a generalized Weibull (GW) distribution is proposed to interpolate between them.