Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations

B. I. Henry, T. A. M. Langlands, and S. L. Wearne
Phys. Rev. E 74, 031116 – Published 14 September 2006

Abstract

We have revisited the problem of anomalously diffusing species, modeled at the mesoscopic level using continuous time random walks, to include linear reaction dynamics. If a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction-diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term. If the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term. Results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term. We have also developed further extensions of the CTRW model to include more general reaction dynamics.

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  • Received 13 April 2006

DOI:https://doi.org/10.1103/PhysRevE.74.031116

©2006 American Physical Society

Authors & Affiliations

B. I. Henry* and T. A. M. Langlands

  • Department of Applied Mathematics, School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia

S. L. Wearne

  • Center for Biomathematical Sciences, Mount Sinai School of Medicine, New York, New York, 10029-6574, USA

  • *Electronic address: B.Henry@unsw.edu.au
  • Electronic address: t.langlands@unsw.edu.au
  • Electronic address: susan.wearne@mssm.edu; Also at Fishberg Department of Neuroscience.; Computational Neurobiology and Imaging Center.

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Vol. 74, Iss. 3 — September 2006

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