Positive time fractional derivative

TL;DR

This paper introduces a positive fractional time derivative to improve the modeling of anomalous diffusions, ensuring the derivative's positivity in Fourier space for fractional or odd integer orders.

Contribution

A new positive fractional time derivative is proposed, addressing the limitations of traditional derivatives in modeling anomalous diffusions.

Findings

The proposed derivative maintains positivity in Fourier transform.

It is applicable to fractional and odd integer orders.

Enhances the mathematical modeling of anomalous diffusions.

Abstract

In mathematical modeling of the non-squared frequency-dependent diffusions, also known as the anomalous diffusions, it is desirable to have a positive real Fourier transform for the time derivative of arbitrary fractional or odd integer order. The Fourier transform of the fractional time derivative in the Riemann-Liouville and Caputo senses, however, involves a complex power function of the fractional order. In this study, a positive time derivative of fractional or odd integer order is introduced to respect the positivity in modeling the anomalous diffusions.