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On a new generalized local fractal derivative operator

Author

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  • El-Nabulsi, Rami Ahmad
  • Khalili Golmankhaneh, Alireza
  • Agarwal, Praveen

Abstract

In this study, a new generalized local fractal derivative operator is introduced and we discuss its implications in classical systems through the Lagrangian and Hamiltonian formalisms. The variational approach has been proved to be practical to describe dissipative dynamical systems. Besides, the Hamiltonian formalism is characterized by the emergence of auxiliary constraints free from Dirac auxiliary functions. In field theory, it was found that both damped Klein-Gordon and Dirac equations are generalized, and for specific parameters, a field equation comparable to the Barut equation describing the electromagnetic interactions between N spin-1/2 particles in lepton physics is obtained. A Hamiltonian formulation of higher-order Lagrangian has been constructed and discussed as well. The reformulation of the problem based on fractal calculus has been also addressed in details and compared with the basic approach.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:chsofr:v:161:y:2022:i:c:s0960077922005392
    DOI: 10.1016/j.chaos.2022.112329
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    References listed on IDEAS

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    1. Kolwankar, Kiran M., 2021. "Exact local fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Dumitru Baleanu & Arran Fernandez, 2019. "On Fractional Operators and Their Classifications," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    3. Z. E. Musielak & N. Davachi & M. Rosario-Franco, 2020. "Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2020, pages 1-11, June.
    4. Musielak, Z.E., 2009. "General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2645-2652.
    5. Johannes Jahn, 2020. "Introduction to the Theory of Nonlinear Optimization," Springer Books, Springer, edition 4, number 978-3-030-42760-3, December.
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    Cited by:

    1. 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).
    2. 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).
    3. 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.
    4. Khalili Golmankhaneh, Alireza & Bongiorno, Donatella, 2024. "Exact solutions of some fractal differential equations," Applied Mathematics and Computation, Elsevier, vol. 472(C).

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