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Generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains.

C N Angstmann1, B I Henry1, A V McGann1

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Summary
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This study introduces a new mathematical model for subdiffusive transport in growing materials using a novel comoving fractional derivative. This framework accurately describes phenomena like protein diffusion in expanding biological membranes.

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Area of Science:

  • Physics
  • Materials Science
  • Biophysics

Background:

  • Subdiffusive transport is common in physical and biological systems.
  • Existing theoretical models often use fractional derivatives but struggle with growing materials.

Purpose of the Study:

  • To develop robust theoretical models for subdiffusive transport in growing media.
  • To introduce a new mathematical framework for these complex processes.

Main Methods:

  • Derivation of evolution equations based on a continuous-time random walk.
  • Introduction of a novel comoving, fractional derivative.

Main Results:

  • Successfully derived evolution equations for subdiffusive transport in growing media.
  • Demonstrated the model's utility with a case study of proteins in a growing membrane.

Conclusions:

  • The new comoving fractional derivative provides a concise and effective method for modeling subdiffusion in growing systems.
  • This work offers a significant advancement for understanding transport phenomena in dynamic materials.