Boundary Conditions for Fractional Diffusion

TL;DR

This paper establishes physically meaningful boundary conditions for fractional diffusion equations, analyzing their properties, and demonstrating the limitations of Caputo derivatives in modeling such processes.

Contribution

It introduces new boundary conditions for fractional diffusion, reviews theoretical properties, and highlights the unsuitability of Caputo derivatives for positivity-preserving models.

Findings

Derived boundary conditions using mass balance

Numerical solutions illustrating boundary effects

Caputo derivative unsuitable for positivity preservation

Abstract

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.