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Fractal calculus approach to diffusion on fractal combs

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  • Khalili Golmankhaneh, Alireza
  • Ontiveros, Lilián Aurora Ochoa

Abstract

In this paper, we present a generalization of diffusion on fractal combs using fractal calculus. We introduce the concept of a fractal comb and its associated staircase function. To handle functions supported on these combs, we define derivatives and integrals using the staircase function. We then derive the Fokker–Planck equation for a fractal comb with dimension α, incorporating fractal time, and provide its solution. Additionally, we explore α-dimensional and (2α)-dimensional Brownian motion on fractal combs with drift and fractal time. We calculate the corresponding fractal mean square displacement for these processes. Furthermore, we propose and solve the heat equation on an α-dimensional fractal comb space. To illustrate our findings, we include graphs that showcase the specific details and outcomes of our results.

Suggested Citation

  • Khalili Golmankhaneh, Alireza & Ontiveros, Lilián Aurora Ochoa, 2023. "Fractal calculus approach to diffusion on fractal combs," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    • Handle: RePEc:eee:chsofr:v:175:y:2023:i:p1:s0960077923009220
    DOI: 10.1016/j.chaos.2023.114021
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    References listed on IDEAS

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    1. Suleiman, Kheder & Song, Qixuan & Zhang, Xuelan & Liu, Shengna & Zheng, Liancun, 2022. "Anomalous diffusion in a circular comb with external velocity field," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    2. Alireza Khalili Golmankhaneh & Renat Timergalievich Sibatov, 2021. "Fractal Stochastic Processes on Thin Cantor-Like Sets," Mathematics, MDPI, vol. 9(6), pages 1-13, March.
    3. Satin, Seema E. & Parvate, Abhay & Gangal, A.D., 2013. "Fokker–Planck equation on fractal curves," Chaos, Solitons & Fractals, Elsevier, vol. 52(C), pages 30-35.
    4. Golmankhaneh, Alireza K. & Tunç, Cemil, 2019. "Sumudu transform in fractal calculus," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 386-401.
    5. Wang, Zhaoyang & Lin, Ping & Wang, Erhui, 2021. "Modeling multiple anomalous diffusion behaviors on comb-like structures," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    6. Sandev, Trifce & Schulz, Alexander & Kantz, Holger & Iomin, Alexander, 2018. "Heterogeneous diffusion in comb and fractal grid structures," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 551-555.
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    Cited by:

    1. Didier Samayoa & Liliana Alvarez-Romero & José Alfredo Jiménez-Bernal & Lucero Damián Adame & Andriy Kryvko & Claudia del C. Gutiérrez-Torres, 2024. "Torricelli’s Law in Fractal Space–Time Continuum," Mathematics, MDPI, vol. 12(13), pages 1-13, June.
    2. Eduardo Reyes de Luna & Andriy Kryvko & Juan B. Pascual-Francisco & Ignacio Hernández & Didier Samayoa, 2024. "Generalized Kelvin–Voigt Creep Model in Fractal Space–Time," Mathematics, MDPI, vol. 12(19), pages 1-13, October.

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