Fractional differential entropy and its application in modeling one-dimensional flow velocity

TL;DR

This paper introduces a fractional differential entropy for continuous distributions, explores its properties, and applies it to model turbulent flow velocity profiles, demonstrating its advantages over existing models.

Contribution

It develops the continuous fractional differential entropy, analyzes its properties, and applies it to model turbulent flow velocities with improved accuracy.

Findings

The fractional differential entropy has unique properties useful for statistical analysis.

The entropy-based model accurately fits turbulent velocity data.

The proposed model outperforms existing entropy-based models in validation.

Abstract

The fractional order generalization of Shannon entropy proposed by Ubriaco has been studied for discrete distributions. In the current paper, we conduct a detailed study of the continuous analogue of this entropy termed as fractional differential entropy and find some interesting properties which makes it stand out among the existing entropies in literature. The studied entropy measure is evaluated analytically and numerically for some well-known continuous distributions, which will be quite useful in reliability analysis works and other statistical studies of complex systems. Further, it has been used to model the one-dimensional vertical velocity profile of turbulent flows in wide open channels. A one-parametric spatial distribution function is utilized for better estimation of the velocity distribution. The validity of the model has been established using experimental and field data through regression analysis. A comparative study is also presented to show the superiority of the proposed model over the existing entropy-based models.