Numerical simulations of anomalous diffusion

TL;DR

This paper develops finite difference and finite element numerical methods to solve fractional order PDEs modeling anomalous diffusion, incorporating Caputo derivatives and boundary conditions, with simulation results demonstrating their effectiveness.

Contribution

It introduces numerical schemes tailored for fractional PDEs with Caputo derivatives, addressing boundary conditions and providing simulation validation.

Findings

Numerical methods successfully simulate anomalous diffusion.

Finite difference and finite element approaches are effective.

Simulation results align with theoretical expectations.

Abstract

In this paper we present numerical methods - finite differences and finite elements - for solution of partial differential equation of fractional order in time for one-dimensional space. This equation describes anomalous diffusion which is a phenomenon connected with the interactions within the complex and non-homogeneous background. In order to consider physical initial-value conditions we use fractional derivative in the Caputo sense. In numerical analysis the boundary conditions of first kind are accounted and in the final part of this paper the result of simulations are presented.