Analytical and Numerical Solution Methods for Some Space-Fractional Reaction–Diffusion Systems via Hankel Transforms

Diffusion within porous media, such as biological tissues, often deviates from conventional Fick’s laws that may be described by space-fractional diffusion equations. Microscale tissue heterogeneity can be represented by the space-fractional Riesz Laplacian operator acting on concentration or, alternatively, by fractional a Riesz gradient of the order β, extending the usual spatial gradient concept. We consider 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 absent but first-order decay takes place. The steady state is derived using Hankel and Mellin transforms, resulting in integral-kernels-containing Bessel functions. We develop and compare three numerical quadrature methods for the Hankel transform: sinc quadrature, Ogata quadrature (based on Bessel zeros), and a hybrid asymptotic–numerical scheme. Numerical results and plots are presented for exponents β=1/2,2/3,3/4 and 1. The integer-order case (β=1) is recovered as a limiting case. The hybrid method is about five times faster than the global quadratures for the same accuracy. The novelty of this work lies in the systematic comparison of numerical methods for this specific class of fractional reaction–diffusion problems.