A space-fractional reaction-diffusion system with cylindrical symmetry
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912699472936960 |
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| author | Prodanov, Dimiter |
| author_facet | Prodanov, Dimiter |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A space-fractional reaction-diffusion system with cylindrical symmetry Prodanov, Dimiter General Mathematics 33F05 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. |
| title | A space-fractional reaction-diffusion system with cylindrical symmetry |
| topic | General Mathematics 33F05 |
| url | https://arxiv.org/abs/2511.07449 |