Finite Domain Anomalous Spreading Consistent with First and Second Law

TL;DR

This paper introduces a new finite domain spatial operator for anomalous diffusion equations that respects physical laws like conservation and the second law of thermodynamics, supported by numerical methods and results.

Contribution

The paper proposes a novel spatial operator for bounded domains in anomalous diffusion models, addressing issues with previous operators and ensuring physical law compliance.

Findings

The new operator satisfies conservation principles.

Numerical methods effectively solve the modified equations.

Results demonstrate adherence to thermodynamic laws.

Abstract

After reviewing the problematic behavior of some previously suggested finite interval spatial operators of the symmetric Riesz type, we create a wish list leading toward a new spatial operator suitable to use in the space-time fractional differential equation of anomalous diffusion when the transport of material is strictly restricted to a bounded domain. Based on recent studies of wall effects, we introduce a new definition of the spatial operator and illustrate its favorable characteristics. We provide two numerical methods to solve the modified space-time fractional differential equation and show particular results illustrating compliance to our established list of requirements, most important to the conservation principle and the second law of thermodynamics.