Work

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

Year: 2026

Type: article

Abstract: 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.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract10080523

Authors: Teodor Vakarelsky, Anish Kumar, Dimiter Prodanov

Institutions: Bulgarian Academy of Sciences, Institute of Information and Communication Technologies

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,258

Funder: European Commission