Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

TL;DR

This paper investigates the mathematical well-posedness of anisotropic nonlocal diffusion equations, including fractional and advection-diffusion types, providing rigorous estimates and extending analysis to anomalous transport phenomena.

Contribution

It establishes equivalence between weighted and unweighted operators and proves well-posedness for fractional orders in anisotropic nonlocal diffusion models.

Findings

Proved well-posedness for fractional orders s in [0.5,1)

Established equivalence between weighted and unweighted operators

Applied results to anomalous solute transport

Abstract

We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s in [0.5,1). We also present an application of the advection-diffusion equation to anomalous transport of solutes.