# Some comments on using fractional derivative operators in modeling non-local diffusion processes

## TL;DR

This paper examines the use of space fractional derivative operators in non-local diffusion models, compares Caputo and Riemann-Liouville flux laws through numerical tests, and proposes a clearer definition to resolve ambiguities.

## Contribution

It introduces a unified flux law definition that clarifies the use of fractional derivatives in non-local diffusion modeling, addressing issues with existing flux laws.

## Key findings

- Riemann-Liouville flux law includes an additional advection-like term.
- RL flux law predictions can be physically and mathematically unsound.
- A new fractional derivative flux law is proposed to eliminate ambiguities.

## Abstract

We start with a general governing equation for diffusion transport, written in a conserved form, in which the phenomenological flux laws can be constructed in a number of alternative ways. We pay particular attention to flux laws that can account for non-locality through space fractional derivative operators. The available results on the well posedness of the governing equations using such flux laws are discussed. A discrete control volume numerical solution of the general conserved governing equation is developed and a general discrete treatment of boundary conditions, independent of the particular choice of flux law, is presented. We use numerical solutions of various test problems to compare the operation and predictive ability of two discrete fractional diffusion flux laws based on the Caputo (C) and Riemann-Liouville (RL) derivatives respectively. When compared with the C flux-law we note that the RL flux law includes an additional term, that, in a phenomenological sense, acts as an apparent advection transport. Through our test solutions we show that, when compared to the performance of the C flux-law, this extra term can lead to RL-flux law predictions that may be physically and mathematically unsound. We conclude, by proposing a parsimonious definition for a fractional derivative based flux law that removes the ambiguities associated with the selection between non-local flux laws based on the RL and C fractional derivatives.