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Updated: Nov 27, 2025

In Situ Monitoring of Diffusion of Guest Molecules in Porous Media Using Electron Paramagnetic Resonance Imaging
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Diffusion on Middle-ξ Cantor Sets.

Alireza Khalili Golmankhaneh1, Arran Fernandez2, Ali Khalili Golmankhaneh3

  • 1Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary

This study introduces a new calculus, Cζ-calculus, for middle-ξ Cantor sets. This fractional calculus framework enables solving differential equations on fractal sets and analyzing diffusion behaviors.

Keywords:
Cζ-calculusHausdorff dimensiondiffusion on fractalmiddle-ξ Cantor setsrandom walkstaircase function

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Area of Science:

  • Fractal Geometry
  • Non-integer Calculus
  • Mathematical Physics

Background:

  • Standard calculus is insufficient for functions with fractal support due to their complex nature.
  • Generalized Cantor sets, specifically middle-ξ Cantor sets, exhibit self-similarity and dimensions exceeding their topological ones.
  • Local fractional derivatives are necessary to analyze functions on fractal supports.

Purpose of the Study:

  • To generalize Cζ-calculus to middle-ξ Cantor sets.
  • To develop a novel calculus applicable to fractal structures.
  • To solve differential equations and analyze diffusion phenomena on these fractal sets.

Main Methods:

  • Generalization of Cζ-calculus for middle-ξ Cantor sets with 0<ξ<1.
  • Application of local fractional derivatives.
  • Solving differential equations on fractal domains.
  • Analysis of super-, normal, and sub-diffusion conditions.

Main Results:

  • A new calculus framework, Cζ-calculus, has been successfully generalized for middle-ξ Cantor sets.
  • Differential equations defined on these fractal sets have been solved.
  • Illustrative examples demonstrate the application and effectiveness of the proposed calculus.
  • Conditions governing different types of diffusion (super-, normal, sub-) on fractal sets are established.

Conclusions:

  • The generalized Cζ-calculus provides a powerful tool for analyzing fractal phenomena.
  • This work extends the applicability of fractional calculus to complex fractal geometries.
  • The findings contribute to understanding diffusion processes on fractal sets.