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A space-fractional reaction-diffusion system with cylindrical symmetry

Dimiter Prodanov

TL;DR

The paper addresses diffusion in porous media with space-fractional diffusion modeled by the Riesz Laplacian in a two-region cylindrical geometry representing a proximal source and an outer decaying region around an implanted electrode. It derives a steady-state formulation, solves the integer-order case analytically via Bessel functions and the fractional-order case via Hankel transforms and asymptotic Fox $H$-function representations. Key contributions include an exact quadrature-based solution, a tractable asymptotic form in terms of the Fox $H$-function, and numerical techniques (Double-Exponential quadrature and Hankel-transform acceleration) to compute concentration profiles. The framework enhances parameter estimation from experimental data and provides a flexible tool for modeling transport in heterogeneous tissues near implants.

Abstract

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.

A space-fractional reaction-diffusion system with cylindrical symmetry

TL;DR

The paper addresses diffusion in porous media with space-fractional diffusion modeled by the Riesz Laplacian in a two-region cylindrical geometry representing a proximal source and an outer decaying region around an implanted electrode. It derives a steady-state formulation, solves the integer-order case analytically via Bessel functions and the fractional-order case via Hankel transforms and asymptotic Fox -function representations. Key contributions include an exact quadrature-based solution, a tractable asymptotic form in terms of the Fox -function, and numerical techniques (Double-Exponential quadrature and Hankel-transform acceleration) to compute concentration profiles. The framework enhances parameter estimation from experimental data and provides a flexible tool for modeling transport in heterogeneous tissues near implants.

Abstract

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.

Paper Structure

This paper contains 17 sections, 52 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The reaction-diffusion system
  3. Integer-order case
  4. Fractional-order case
  5. An asymptotic solution in terms of H-functions
  6. Numerical experiments
  7. The Double-Exponential quadrature method
  8. Convergence acceleration method for numerical Hankel transforms
  9. Plots of the solutions
  10. Discussion
  11. Integral Transforms
  12. Fourier transform
  13. Hankel transform convention
  14. Mellin transform
  15. The Riesz Laplacian operator
  16. ...and 2 more sections

Figures (6)

  • Figure 1: Geometry of the reaction-diffusion system
  • Figure 2: Comparison of the integer-order solutions
  • Figure 3: Comparison of the fractional-order solutions
  • Figure 4: Influence of the fractional exponent on the shape of the full solution
  • Figure 5: Asymptotic behavior of $c_a(z)$ for $\alpha=0.995$
  • ...and 1 more figures