Numerical solution for a non-Fickian diffusion in a periodic potential

TL;DR

This paper presents a numerical method for solving a hyperbolic non-Fickian diffusion equation in a periodic potential, analyzing convergence and providing results on particle behavior in different regimes.

Contribution

It introduces a Laplace transform-based numerical approach for non-Fickian diffusion in periodic potentials, including convergence analysis and application to inertial and diffusive regimes.

Findings

Numerical method effectively solves the hyperbolic diffusion equation.

Results demonstrate accurate particle density, flux, and mean-square-displacement.

Method captures both inertial and diffusive particle behaviors.

Abstract

Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and mean-square-displacement (covering both inertial and diffusive regimes) are presented.