Numerical solution for a non-Fickian diffusion in a periodic potential
This provides a numerical method for a specific physical model, but the approach (Laplace transform + finite differences) is standard and the results are incremental.
The paper numerically solves a non-Fickian (hyperbolic) diffusion equation for a Brownian particle in a symmetric periodic potential, presenting results for density, flux, and mean-square-displacement across inertial and diffusive regimes.
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.