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This study explores variable-order time-fractional diffusion equations derived from random walk models. Researchers analyzed subdiffusive systems and validated continuous models against lattice-based simulations.

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Area of Science:

  • Physics
  • Mathematics
  • Computational Science

Background:

  • Continuous time random walk (CTRW) models describe particle transport.
  • Fractional calculus offers advanced tools for modeling anomalous diffusion.

Purpose of the Study:

  • Derive and solve variable-order time-fractional diffusion equations.
  • Analyze subdiffusive systems with position-dependent parameters.
  • Compare lattice-based simulations with continuous models.

Main Methods:

  • Developed integrodifferential equations from lattice CTRW schemes.
  • Focused on subdiffusive cases with power-law waiting times.
  • Employed Laplace domain analysis and Gaver-Stehfest algorithm for numerical inversion.

Main Results:

  • Investigated two systems: two-part subdiffusion and linearly changing subdiffusion exponent.
  • Validated numerical solutions of generalized master equations against continuous equations.
  • Demonstrated the applicability of fractional diffusion equations to complex systems.

Conclusions:

  • Variable-order time-fractional diffusion equations accurately model complex subdiffusive phenomena.
  • Continuous models provide a valid approximation for lattice-based random walk schemes.
  • The methods are applicable to systems with spatially varying anomalous diffusion properties.