Analysis and numerics of traveling waves for asymmetric fractional reaction-diffusion equations

TL;DR

This paper studies traveling wave solutions in a one-dimensional reaction-diffusion system with asymmetric fractional diffusion, providing analytical proofs of existence, uniqueness, stability, and numerical visualization methods.

Contribution

It offers new analytical results on traveling waves for reaction-diffusion equations with Riesz-Feller operators and introduces a numerical discretization approach for visualization.

Findings

Existence and uniqueness of traveling waves established.

Stability of the traveling wave solutions demonstrated.

Numerical methods for visualizing solutions developed.

Abstract

We consider a scalar reaction-diffusion equation in one spatial dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. We present our analytical results on the existence, uniqueness (up to translations) and stability of a traveling wave solution connecting two stable homogeneous steady states. Moreover, we review numerical methods for the case of reaction-diffusion equations with fractional Laplacian and discuss possible extensions to our reaction-diffusion equations with Riesz-Feller operators. In particular, we present a direct method using integral operator discretization in combination with projection boundary conditions to visualize our analytical results about traveling waves.