Numerical and Analytical Solutions for 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 a space-fractional Riesz Laplacian operator acting on the concentration. 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 an 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 fractional exponents β=1/2,2/3,3/4 (2α=1+β). The integer-order case (α=1) is recovered as a limiting case. The hybrid method is about five times faster than the global quadratures for the same accuracy.