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Moving boundary problems governed by anomalous diffusion.

Christopher J Vogl1, Michael J Miksis, Stephen H Davis

  • 1Department of Engineering Sciences and Applied Mathematics , Northwestern University , 2145 Sheridan Road, Evanston, IL 60208-3125, USA.

Proceedings. Mathematical, Physical, and Engineering Sciences
|December 1, 2012
PubMed
Summary
This summary is machine-generated.

This study introduces a new numerical method for anomalous diffusion problems. It reveals that interfaces in subdiffusion systems exhibit non-monotone behavior, initially advancing before receding.

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Area of Science:

  • Mathematical Physics
  • Computational Science
  • Non-equilibrium Systems

Background:

  • Anomalous diffusion, characterized by mean-squared displacement proportional to t(α) with α≠1, deviates from classical diffusion (α=1).
  • Subdiffusion (α<1) presents unique challenges due to its singular history kernel, complicating numerical simulations.
  • Moving boundary problems are crucial for modeling various physical phenomena but are complex when incorporating anomalous diffusion.

Purpose of the Study:

  • To develop a novel numerical method for one-dimensional moving boundary problems involving anomalous diffusion, specifically subdiffusion.
  • To analyze the behavior of interfaces in systems with subdiffusion regions adjacent to classical diffusion or other subdiffusion regions.
  • To investigate the non-monotone interface dynamics arising from subdiffusion phenomena.

Main Methods:

  • Development of a novel numerical technique capable of handling both moving interfaces and the singular history kernel of subdiffusion.
  • Application of the numerical method to two distinct one-dimensional moving boundary problems.
  • Simulation of interface dynamics in scenarios involving subdiffusion adjacent to classical diffusion and subdiffusion adjacent to subdiffusion.

Main Results:

  • The developed numerical method successfully simulates moving boundary problems with subdiffusion.
  • In a subdiffusion-classical diffusion system, the interface exhibits non-monotone behavior, initially advancing and then receding.
  • In a subdiffusion-subdiffusion system, the interface also reverses direction, with the region of higher subdiffusivity initially advancing before receding.

Conclusions:

  • The study presents a robust numerical approach for tackling complex anomalous diffusion problems.
  • Anomalous diffusion, particularly subdiffusion, leads to counterintuitive interface dynamics, including direction reversal.
  • The findings have implications for understanding transport phenomena in complex media where anomalous diffusion is prevalent.