A general definition of the fractal derivative: Theory and applications

Lakhlifa Sadek; Ahmad Sami Bataineh; El Mostafa Sadek; Ishak Hashim; Lakhlifa Sadek; Ahmad Sami Bataineh; El Mostafa Sadek; Ishak Hashim

doi:10.3934/math.2025690

AIMS Mathematics

2025, Volume 10, Issue 7: 15390-15409. doi: 10.3934/math.2025690

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Research article

A general definition of the fractal derivative: Theory and applications

1.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamilnadu, India
2.
Department of Mathematics, Faculty of Sciences and Technology, BP 34. Ajdir 32003 Al-Hoceima, Abdelmalek Essaadi University, Tetouan, Morocco
3.
Department of Mathematics, Faculty of Science, Al-Balqa Applied University, 19117 Al Salt, Jordan
4.
Laboratory of Engineering Sciences for Energy, National School of Applied Sciences of El Jadida, University Chouaib Doukkali, El Jadida 24000, Morocco
5.
Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), Bangi 43650, Selangor, Malaysia
6.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman P.O. Box 346, United Arab Emirates

Received: 10 December 2024 Revised: 18 June 2025 Accepted: 23 June 2025 Published: 03 July 2025
MSC : 26A33

In this paper, we introduce a general definition of the fractal derivative with respect to a function $ \psi $, in the context of the order $ 0 < \alpha\leq 1 $ and the function $ \psi(\Theta) $. This novel definition generalizes the classical fractal derivative, which is recovered when $ \psi(\Theta) = \Theta $, as described in previous works by Chen et al. ^[1,2]. We explored key properties of the $ \psi $-fractal derivative, including the $ \psi $-fractal Laplace transform, which provides a powerful tool for solving complex differential equations in fractal domains. We also derived a generalized $ \psi $-chain rule, extending classical calculus into the fractal domain, and presented fundamental operations related to this unique derivative. We give some applications.
- fractional derivative,
- $ \psi $-fractal derivative,
- $ \psi $-chain rule,
- $ \psi $-fractal integral,
- $ \psi $-fractal Laplace transform
Citation: Lakhlifa Sadek, Ahmad Sami Bataineh, El Mostafa Sadek, Ishak Hashim. A general definition of the fractal derivative: Theory and applications[J]. AIMS Mathematics, 2025, 10(7): 15390-15409. doi: 10.3934/math.2025690

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Abstract

In this paper, we introduce a general definition of the fractal derivative with respect to a function $ \psi $, in the context of the order $ 0 < \alpha\leq 1 $ and the function $ \psi(\Theta) $. This novel definition generalizes the classical fractal derivative, which is recovered when $ \psi(\Theta) = \Theta $, as described in previous works by Chen et al. ^[1,2]. We explored key properties of the $ \psi $-fractal derivative, including the $ \psi $-fractal Laplace transform, which provides a powerful tool for solving complex differential equations in fractal domains. We also derived a generalized $ \psi $-chain rule, extending classical calculus into the fractal domain, and presented fundamental operations related to this unique derivative. We give some applications.

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