Comparative Numerical Analysis of Bessel Integrals Arising in Space-Fractional Diffusion

Diffusion within porous biological media, such as brain tissue, often exhibits anomalous behaviour that deviates from classical Fickian laws, motivating the use of space-fractional diffusion models. Analytically tractable space-fractional diffusion problems are typically solved using an integral transform. Bessel integrals arise in the scope of the radial Fourier transform on the plane, which is equivalent to the Hankel transform, and suffer from slow convergence.

This study focuses on the numerical evaluation of a space-fractional reaction-diffusion system with cylindrical symmetry that models the foreign body reaction around an implanted neural electrode. The system incorporates a space-fractional Riesz Laplacian to capture microscopic tissue heterogeneity, and its steady-state solution is derived via the Hankel transform, yielding expressions involving oscillatory Bessel integrals.

Numerical evaluation of these integrals is essential for practical application and parameter estimation. The foundational work implemented a Double-Exponential (DE) quadrature method, enhanced by a sine hyperbolic transformation for oscillatory kernels and accelerated through Wynn’s epsilon algorithm applied over intervals partitioned at Bessel function zeros. In this extended analysis, we compare the initial DE approach with two additional advanced quadrature methods: the Ogata quadrature and the sinc integration rule. Each method is systematically assessed for accuracy, convergence rate, and computational efficiency across a range of fractional orders and spatial ranges.

Our results demonstrate that fractional exponents produce heavy-tailed concentration profiles distinct from the exponential decay of integer-order solutions, and they provide clear performance benchmarks for these quadrature techniques. This comparative study not only validates the robustness of the numerical framework but also offers practical guidance for selecting efficient integration strategies when calibrating transport parameters from experimental data, thereby enhancing the reliability of anomalous diffusion models in biomedical applications.