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

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  • 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.

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  • 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
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    1. Lacan, Francis & Tresser, Charles, 2015. "Fractals as objects with nontrivial structures at all scales," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 218-242.
    2. Chen, Wen & Liang, Yingjie, 2017. "New methodologies in fractional and fractal derivatives modeling," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 72-77.
    3. 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).
    4. 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).
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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).
    12. 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.
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