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Behavior of fractional diffusion at the origin.

Ya E Ryabov1

  • 1Department of Applied Physics, School of Applied Science, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel. Ryabov@vms.huji.ac.il

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 4, 2003
PubMed
Summary

The Riemann-Liouville fractional time derivative in diffusion equations causes a concentration divergence at the origin, implying an external source. Without this constraint, the model shows no divergence and normal decay.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Chemical Engineering

Background:

  • Fractional diffusion equations model anomalous transport phenomena.
  • Riemann-Liouville fractional time derivatives are used to describe non-local time dependencies in diffusion processes.

Purpose of the Study:

  • To analyze the implications of the normalization conservation constraint in fractional diffusion equations using Riemann-Liouville fractional time derivatives.
  • To investigate the behavior of diffusive agent concentration at the origin under different constraint conditions.

Main Methods:

  • Mathematical modeling using fractional diffusion equations.
  • Analysis of the Riemann-Liouville fractional time derivative.
  • Investigating the behavior of the diffusive agent concentration at the origin (r-->0).

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Main Results:

  • The normalization conservation constraint leads to a divergence in diffusive agent concentration at the origin.
  • This divergence suggests the presence of an external source of the diffusive agent at r-->0.
  • Absence of the normalization conservation constraint prevents divergence and ensures normal decay of the diffusive agent concentration.

Conclusions:

  • The Riemann-Liouville fractional time derivative, under normalization conservation, implies a loss of diffusive agent mass compensated by an origin source.
  • The model without the normalization conservation constraint provides a more physically plausible scenario without unphysical divergences.