IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v168y2023ics0960077923001042.html
   My bibliography  Save this article

Vector differential operators in a fractional dimensional space, on fractals, and in fractal continua

Author

Listed:
  • Balankin, Alexander S.
  • Mena, Baltasar

Abstract

This paper is devoted to the development of local vector calculus in fractional-dimensional spaces, on fractals, and in fractal continua. We conjecture that in the space of non-integer dimension one can define two different del-operators acting on the scalar and vector fields respectively. The basic vector differential operators and Laplacian in the fractional-dimensional space are expressed in terms of two del-operators in a conventional way. Likewise, we construct Laplacian and vector differential operators associated with Fα-derivatives on fractals. The conjugacy between Fα and ordinary derivatives allow us to map the vector differential operators on the fractal domain onto the vector differential calculus in the corresponding fractal continuum. These results provide a novel tool for modeling physical phenomena in complex systems.

Suggested Citation

  • Balankin, Alexander S. & Mena, Baltasar, 2023. "Vector differential operators in a fractional dimensional space, on fractals, and in fractal continua," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
  • Handle: RePEc:eee:chsofr:v:168:y:2023:i:c:s0960077923001042
    DOI: 10.1016/j.chaos.2023.113203
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077923001042
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2023.113203?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. Qaisar Abbas Naqvi & Muhammad Arshad Fiaz, 2017. "Electromagnetic behavior of a planar interface of non-integer dimensional spaces," Journal of Electromagnetic Waves and Applications, Taylor & Francis Journals, vol. 31(16), pages 1625-1637, November.
    2. Lacan, Francis & Tresser, Charles, 2015. "Fractals as objects with nontrivial structures at all scales," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 218-242.
    3. Chen, Wen & Liang, Yingjie, 2017. "New methodologies in fractional and fractal derivatives modeling," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 72-77.
    4. Weberszpil, J. & Helayël-Neto, J.A., 2016. "Variational approach and deformed derivatives," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 450(C), pages 217-227.
    5. Borges, Ernesto P., 2004. "A possible deformed algebra and calculus inspired in nonextensive thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 95-101.
    6. Weberszpil, J. & Lazo, Matheus Jatkoske & Helayël-Neto, J.A., 2015. "On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 399-404.
    7. El-Nabulsi, Rami Ahmad & Khalili Golmankhaneh, Alireza & Agarwal, Praveen, 2022. "On a new generalized local fractal derivative operator," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    8. Balankin, Alexander S. & Ramírez-Joachin, Juan & González-López, Gabriela & Gutíerrez-Hernández, Sebastián, 2022. "Formation factors for a class of deterministic models of pre-fractal pore-fracture networks," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    9. Alexander S. Balankin & Juliã N Patiã‘O Ortiz & Miguel Patiã‘O Ortiz, 2022. "Inherent Features Of Fractal Sets And Key Attributes Of Fractal Models," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(04), pages 1-23, June.
    10. Balankin, Alexander S., 2020. "Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    11. Rosa, Wanderson & Weberszpil, José, 2018. "Dual conformable derivative: Definition, simple properties and perspectives for applications," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 137-141.
    12. Balankin, Alexander S. & Bory-Reyes, Juan & Shapiro, Michael, 2016. "Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 345-359.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Li, Peiluan & Han, Liqin & Xu, Changjin & Peng, Xueqing & Rahman, Mati ur & Shi, Sairu, 2023. "Dynamical properties of a meminductor chaotic system with fractal–fractional power law operator," Chaos, Solitons & Fractals, Elsevier, vol. 175(P2).
    2. Zhang, Shuai & Li, Yingjun & Wang, Guicong & Qi, Zhenguang & Zhou, Yuanqin, 2024. "A novel method for calculating the fractal dimension of three-dimensional surface topography on machined surfaces," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    3. 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.
    4. Didier Samayoa & Alexandro Alcántara & Helvio Mollinedo & Francisco Javier Barrera-Lao & Christopher René Torres-SanMiguel, 2023. "Fractal Continuum Mapping Applied to Timoshenko Beams," Mathematics, MDPI, vol. 11(16), pages 1-12, August.
    5. Khalili Golmankhaneh, Alireza & Bongiorno, Donatella, 2024. "Exact solutions of some fractal differential equations," Applied Mathematics and Computation, Elsevier, vol. 472(C).

    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. Balankin, Alexander S., 2020. "Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    2. 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.
    3. Goulart, A.G. & Lazo, M.J. & Suarez, J.M.S., 2020. "A deformed derivative model for turbulent diffusion of contaminants in the atmosphere," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    4. Rosa, Wanderson & Weberszpil, José, 2018. "Dual conformable derivative: Definition, simple properties and perspectives for applications," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 137-141.
    5. Qiu, Lin & Lin, Ji & Chen, Wen & Wang, Fajie & Hua, Qingsong, 2020. "A novel method for image edge extraction based on the Hausdorff derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    6. Li, Peiluan & Han, Liqin & Xu, Changjin & Peng, Xueqing & Rahman, Mati ur & Shi, Sairu, 2023. "Dynamical properties of a meminductor chaotic system with fractal–fractional power law operator," Chaos, Solitons & Fractals, Elsevier, vol. 175(P2).
    7. Umpierrez, Haridas & Davis, Sergio, 2021. "Fluctuation theorems in q-canonical ensembles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).
    8. Khalili Golmankhaneh, Alireza & Bongiorno, Donatella, 2024. "Exact solutions of some fractal differential equations," Applied Mathematics and Computation, Elsevier, vol. 472(C).
    9. Balankin, Alexander S. & Ramírez-Joachin, Juan & González-López, Gabriela & Gutíerrez-Hernández, Sebastián, 2022. "Formation factors for a class of deterministic models of pre-fractal pore-fracture networks," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    10. Kin Keung Lai & Shashi Kant Mishra & Ravina Sharma & Manjari Sharma & Bhagwat Ram, 2023. "A Modified q-BFGS Algorithm for Unconstrained Optimization," Mathematics, MDPI, vol. 11(6), pages 1-24, March.
    11. Zhokh, Alexey & Strizhak, Peter, 2018. "Thiele modulus having regard to the anomalous diffusion in a catalyst pellet," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 58-63.
    12. Megías, E. & Timóteo, V.S. & Gammal, A. & Deppman, A., 2022. "Bose–Einstein condensation and non-extensive statistics for finite systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 585(C).
    13. Martinez, Alexandre Souto & González, Rodrigo Silva & Terçariol, César Augusto Sangaletti, 2008. "Continuous growth models in terms of generalized logarithm and exponential functions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(23), pages 5679-5687.
    14. Zine El Abiddine Fellah & Mohamed Fellah & Nicholas O. Ongwen & Erick Ogam & Claude Depollier, 2021. "Acoustics of Fractal Porous Material and Fractional Calculus," Mathematics, MDPI, vol. 9(15), pages 1-16, July.
    15. Chen, Wen & Liang, Yingjie, 2017. "New methodologies in fractional and fractal derivatives modeling," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 72-77.
    16. Golmankhaneh, Alireza K. & Tunç, Cemil, 2019. "Sumudu transform in fractal calculus," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 386-401.
    17. Marco A. S. Trindade & Sergio Floquet & Lourival M. S. Filho, 2018. "Portfolio Theory, Information Theory and Tsallis Statistics," Papers 1811.07237, arXiv.org, revised Oct 2019.
    18. Serkan Araci & Gauhar Rahman & Abdul Ghaffar & Azeema & Kottakkaran Sooppy Nisar, 2019. "Fractional Calculus of Extended Mittag-Leffler Function and Its Applications to Statistical Distribution," Mathematics, MDPI, vol. 7(3), pages 1-14, March.
    19. Oikonomou, Th., 2007. "Tsallis, Rényi and nonextensive Gaussian entropy derived from the respective multinomial coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(1), pages 119-134.
    20. Agathiyan, A. & Gowrisankar, A. & Fataf, Nur Aisyah Abdul, 2024. "On the integral transform of fractal interpolation functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 222(C), pages 209-224.

    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:168:y:2023:i:c:s0960077923001042. 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.