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Time-fractional diffusion equation with time dependent diffusion coefficient.

Kwok Sau Fa1, E K Lenzi

  • 1Departamento de Física, Universidade Estadual de Maringá, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2005
PubMed
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This study analyzes the time-fractional diffusion equation with a time-dependent diffusion coefficient. Results show mean squared displacement scales with t^gamma, and mean first passage time is finite in superdiffusive regimes.

Area of Science:

  • Mathematical Physics
  • Partial Differential Equations
  • Anomalous Diffusion

Background:

  • Fractional diffusion equations model anomalous transport phenomena.
  • Time-dependent coefficients introduce complexities in analyzing diffusion processes.

Purpose of the Study:

  • To investigate solutions of the time-fractional diffusion equation with a time-dependent diffusion coefficient.
  • To analyze mean squared displacement and mean first passage time for different regimes.

Main Methods:

  • Utilizing the Caputo time-fractional derivative operator.
  • Solving the fractional diffusion equation in infinite and finite domains.
  • Deriving expressions for mean squared displacement and mean first passage time.

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Main Results:

  • The mean squared displacement is found to be approximately t^gamma for alpha = 0.
  • The mean first passage time is confirmed to be finite for superdiffusive scenarios.
  • Solutions are analyzed for both infinite and finite domain cases.

Conclusions:

  • The time-fractional diffusion equation with a time-dependent coefficient exhibits distinct behaviors in different diffusion regimes.
  • The derived results provide insights into anomalous transport phenomena.
  • The study contributes to the understanding of fractional calculus applications in physics.