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Area of Science:

  • Mathematical Physics
  • Statistical Mechanics
  • Non-equilibrium Systems

Background:

  • Generalized Fokker-Planck and diffusion-wave equations model complex systems with memory effects.
  • Standard subordination is a key technique for analyzing stochastic processes.

Purpose of the Study:

  • To apply the evolution operator method to solve generalized Fokker-Planck and diffusion-wave equations.
  • To establish an analogy between the evolution operator method and standard subordination.
  • To investigate the role of memory functions and initial conditions.

Main Methods:

  • The evolution operator method is employed.
  • Analysis involves memory functions represented as integral kernels.
  • Laplace transforms are used to analyze memory functions.
  • Power-law memory functions are considered.

Main Results:

  • The method yields probability density functions analogous to standard subordination for generalized Fokker-Planck equations.
  • Diffusion-like initial conditions are required for generalized diffusion-wave equations to achieve this analogy.
  • Power-law memory functions lead to characterization by one-sided stable Lévy distributions.
  • Properties of evolution operators, including evolution and self-reproduction, are examined.

Conclusions:

  • The evolution operator method provides a robust framework for solving generalized diffusion equations with memory.
  • The analogy to subordination is confirmed, offering deeper insights into the underlying stochastic processes.
  • The choice of memory functions and initial conditions is crucial for the behavior of the system.