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This study introduces distributed-order time fractional diffusion equations with multifractal memory kernels, offering a novel approach to anomalous diffusion modeling beyond simple power-law behaviors.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Complex Systems

Background:

  • Anomalous diffusion is often modeled using fractional diffusion equations with power-law memory kernels.
  • The continuous time random walk (CTRW) model by Scher, Montroll, and Weiss provides a physical basis for anomalous diffusion.
  • Distributed-order fractional calculus offers a more generalized framework for describing memory effects.

Purpose of the Study:

  • To analyze distributed-order time fractional diffusion equations featuring multifractal memory kernels.
  • To compare natural and modified-form distributed-order diffusion equations.
  • To establish connections between standard diffusion processes and these generalized fractional models.

Main Methods:

  • Analysis of distributed-order time fractional diffusion equations.
  • Application of the continuous time random walk (CTRW) physical approach.
  • Derivation of the mean squared displacement and analysis of its limiting behavior.
  • Development of a generalized subordination of time to link Wiener processes with fractional dynamics.

Main Results:

  • Characterization of distributed-order time fractional diffusion equations with multifractal memory kernels.
  • Obtained and analyzed the mean squared displacement for these novel equations.
  • Established a generalized subordination of time connecting conventional Langevin dynamics (Wiener process) to the studied fractional diffusion equations.
  • Provided a detailed analysis of the multifractal properties inherent in these distributed-order diffusion models.

Conclusions:

  • Distributed-order time fractional diffusion equations with multifractal memory kernels provide a richer description of anomalous diffusion.
  • The generalized subordination of time offers a powerful tool for understanding the relationship between classical and fractional diffusion dynamics.
  • This work advances the understanding of complex memory effects in anomalous diffusion processes.