# The role of fractional time-derivative operators on anomalous diffusion

## TL;DR

This paper explores how new fractional time-derivative operators with non-singular kernels can model complex anomalous diffusion processes, revealing diverse diffusive behaviors, non-Gaussian distributions, and connections to stochastic resetting.

## Contribution

It introduces and analyzes the use of Caputo-Fabrizio and Atangana-Baleanu fractional operators in diffusion equations, expanding modeling capabilities for anomalous diffusion.

## Key findings

- Derived exact probability distributions for various diffusion regimes
- Identified crossovers between different diffusive behaviors
- Linked new fractional operators to stochastic resetting processes

## Abstract

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results show that these new operators are a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.