2025/11/05 by Dimiter Prodanov, Prodanov, Dimiter
Engineering · Mathematics · Physics and Astronomy · #33F05 #Advanced Control Systems Design #FOS: Mathematics #Fractional Differential Equations Solutions #General Mathematics (math.GM) #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.2511.07449
openalex publication_date 2025/11/05 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28
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.