Numerical solutions to boundary value problem for anomalous diffusion equation with Riesz-Feller fractional operator

TL;DR

This paper develops a finite difference numerical method to solve boundary value problems involving Riesz-Feller fractional derivatives, modeling anomalous diffusion with applications to temperature profiles in nanotubes.

Contribution

It introduces a novel finite difference approach for solving fractional boundary value problems with Riesz-Feller derivatives, applicable to complex diffusion processes.

Findings

Numerical method effectively models anomalous diffusion.

Simulation results demonstrate temperature profiles in nanotubes.

Method accurately approximates solutions to fractional differential equations.

Abstract

In this paper, we present a numerical solution to an ordinary differential equation of a fractional order in one-dimensional space. The solution to this equation can describe a steady state of the process of anomalous diffusion. The process arises from interactions within complex and non-homogeneous background. We present a numerical method which is based on the finite differences method. We consider a boundary value problem (Dirichlet conditions) for an equation with the Riesz-Feller fractional derivative. In the final part of this paper, same simulation results are shown. We present an example of non-linear temperature profiles in nanotubes which can be approximated by a solution to the fractional differential equation.