IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v175y2023ip1s0960077923009220.html
   My bibliography  Save this article

Fractal calculus approach to diffusion on fractal combs

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077923009220
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2023.114021?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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).
    4. Alireza Khalili Golmankhaneh & Renat Timergalievich Sibatov, 2021. "Fractal Stochastic Processes on Thin Cantor-Like Sets," Mathematics, MDPI, vol. 9(6), pages 1-13, March.
    5. Golmankhaneh, Alireza K. & Tunç, Cemil, 2019. "Sumudu transform in fractal calculus," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 386-401.
    6. Wang, Zhaoyang & Lin, Ping & Wang, Erhui, 2021. "Modeling multiple anomalous diffusion behaviors on comb-like structures," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Liu, Lin & Chen, Siyu & Bao, Chunxu & Feng, Libo & Zheng, Liancun & Zhu, Jing & Zhang, Jiangshan, 2023. "Analysis of the absorbing boundary conditions for anomalous diffusion in comb model with Cattaneo model in an unbounded region," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    2. Ayse Gertik & Aykut Karaman, 2023. "The Fractal Approach in the Biomimetic Urban Design: Le Corbusier and Patrick Schumacher," Sustainability, MDPI, vol. 15(9), pages 1-21, May.
    3. Wang, Zhaoyang & Zheng, Liancun, 2020. "Anomalous diffusion in inclined comb-branch structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 549(C).
    4. Guo, Wei & Liu, Ying-Zhou & Huang, Fei-Jie & Shi, Hong-Da & Du, Lu-Chun, 2023. "Brownian particles in a periodic potential corrugated by disorder: Anomalous diffusion and ergodicity breaking," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    5. Golmankhaneh, Alireza K. & Tunc, Cemil, 2017. "On the Lipschitz condition in the fractal calculus," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 140-147.
    6. Tawfik, Ashraf M. & Elkamash, I.S., 2022. "On the correlation between Kappa and Lévy stable distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    7. Zhang, Zhiguo & Kon, Mark A., 2022. "Wavelet matrix operations and quantum transforms," Applied Mathematics and Computation, Elsevier, vol. 428(C).
    8. Takahashi, Hiroshi & Tamura, Yozo, 2023. "Diffusion processes in Brownian environments on disconnected selfsimilar fractal sets in R," Statistics & Probability Letters, Elsevier, vol. 193(C).
    9. Satin, Seema & Gangal, A.D., 2019. "Random walk and broad distributions on fractal curves," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 17-23.
    10. Wu, Junru, 2020. "On a linearity between fractal dimension and order of fractional calculus in Hölder space," Applied Mathematics and Computation, Elsevier, vol. 385(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:175:y:2023:i:p1:s0960077923009220. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.