First-Order Reaction-Diffusion System with Space-Fractional Diffusion in an Unbounded Medium

Abstract

Diffusion in porous media, such as biological tissues, is characterized by deviations from the usual Fick's diffusion laws, which can lead to space-fractional diffusion. The paper considers a simple case of a reaction-diffusion system with two spatial compartments - a proximal one of finite width having a source; and a distal one, which is extended to infinity and where the source is not present but there is a first order decay of the diffusing species. The diffusion term is taken to be proportional to the Riesz fractional Laplacian operator. It is demonstrated that the steady state of the system can be solved in terms of the Hankel transform involving Bessel functions. Methods for numerical evaluation of the resulting integrals are implemented. It is demonstrated that the convergence of the Bessel integrals could be accelerated using standard techniques for sequence acceleration.