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Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric

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  • Balankin, Alexander S.
  • Bory-Reyes, Juan
  • Shapiro, Michael

Abstract

One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil–Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:444:y:2016:i:c:p:345-359
    DOI: 10.1016/j.physa.2015.10.035
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    References listed on IDEAS

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    1. Rufeil Fiori, E. & Plastino, A., 2013. "A Shannon–Tsallis transformation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(8), pages 1742-1749.
    2. Kalogeropoulos, Nikos, 2012. "Tsallis entropy induced metrics and CAT(k) spaces," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(12), pages 3435-3445.
    3. Plastino, A. & Rocca, M.C., 2013. "The Tsallis–Laplace transform," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(22), pages 5581-5591.
    4. 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.
    5. Piasecki, R. & Martin, M.T. & Plastino, A., 2002. "Inhomogeneity and complexity measures for spatial patterns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 307(1), pages 157-171.
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    Citations

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    Cited by:

    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. 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.
    3. 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).
    4. 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).
    5. 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).
    6. Qadeer, Neelam & Bhatti, Nayab & Naqvi, Qaisar Abbas & Fiaz, Muhammad Arshad, 2019. "Use of Kobayashi potential method and Lorentz–Drude model to study scattering from a PEC strip buried below a lossy dispersive NID dielectric-magnetic slab," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.

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