A space-fractional reaction-diffusion system with cylindrical symmetry

Abstract

Diffusion within porous media, such as biological tissues, exhibits departures from conventional Fick's laws, which could result in space-fractional diffusion. The paper considers a reaction-diffusion system with two spatial compartments -- a proximal one of finite radius having a source, and an outer one extending to infinity where the source is not present but first-order decay of the diffusing species takes place. The system models the foreign body reaction around an implanted electrode. Microscopic heterogeneity inside the tissue was modeled by a space-fractional Riesz Laplacian acting on the concentration. This allows for a flexible approach when estimating transport parameters from experimental data. The steady-state of the system is solved in terms of Hankel and Mellin transforms, resulting in a Fox H-function. In the integer-order case, the analytical solution reduces to a superposition of modified Bessel functions of the first and second kinds. Solutions are exhibited by numerical quadrature of the involved Bessel function integrals.