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Fractional calculus and the evolution of fractal phenomena

Author

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  • Rocco, Andrea
  • West, Bruce J.

Abstract

It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we demonstrate that the fractional derivative (integral) of a generalized Weierstrass function (GWF) is another fractal function with a greater (lesser) fractal dimension. We also determine that the GWF is a solution to such a fractional differential stochastic equation of motion.

Suggested Citation

  • Rocco, Andrea & West, Bruce J., 1999. "Fractional calculus and the evolution of fractal phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 265(3), pages 535-546.
    • Handle: RePEc:eee:phsmap:v:265:y:1999:i:3:p:535-546
    DOI: 10.1016/S0378-4371(98)00550-0
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    Cited by:

    1. Ahmed, E. & Elgazzar, A.S., 2007. "On fractional order differential equations model for nonlocal epidemics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(2), pages 607-614.
    2. Brechtl, Jamieson & Xie, Xie & Liaw, Peter K. & Zinkle, Steven J., 2018. "Complexity modeling and analysis of chaos and other fluctuating phenomena," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 166-175.
    3. Ngueuteu Mbouna, S.G. & Banerjee, Tanmoy & Yamapi, René & Woafo, Paul, 2022. "Diverse chimera and symmetry-breaking patterns induced by fractional derivation effect in a network of Stuart-Landau oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    4. Debbouche, Nadjette & Almatroud, A. Othman & Ouannas, Adel & Batiha, Iqbal M., 2021. "Chaos and coexisting attractors in glucose-insulin regulatory system with incommensurate fractional-order derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    5. Alireza Khalili Golmankhaneh & Renat Timergalievich Sibatov, 2021. "Fractal Stochastic Processes on Thin Cantor-Like Sets," Mathematics, MDPI, vol. 9(6), pages 1-13, March.
    6. Wang, Peng & Huo, Jie & Wang, Xu-Ming & Wang, Bing-Hong, 2022. "Diffusion and memory effect in a stochastic process and the correspondence to an information propagation in a social system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 607(C).
    7. Wang, Yupin & Liu, Shutang & Li, Hui & Wang, Da, 2019. "On the spatial Julia set generated by fractional Lotka-Volterra system with noise," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 129-138.

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