Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus
(2023) In Entropy 25(7).- Abstract
This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff’s concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo’s derivative is also proportional to a continuous... (More)
This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff’s concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo’s derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry.
(Less)
- author
- Deppman, Airton ; Megías, Eugenio and Pasechnik, Roman LU
- organization
- publishing date
- 2023-07
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- fractal derivatives, fractional derivatives, fractional differential equations, nonextensive statistics, q-calculus
- in
- Entropy
- volume
- 25
- issue
- 7
- article number
- 1008
- publisher
- MDPI AG
- external identifiers
-
- scopus:85173704979
- pmid:37509954
- ISSN
- 1099-4300
- DOI
- 10.3390/e25071008
- language
- English
- LU publication?
- yes
- additional info
- Publisher Copyright: © 2023 by the authors.
- id
- f8b4dca9-7ab5-4982-a9b8-fbfda4402a3a
- date added to LUP
- 2023-11-10 14:00:43
- date last changed
- 2025-07-03 11:36:29
@article{f8b4dca9-7ab5-4982-a9b8-fbfda4402a3a,
abstract = {{<p>This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff’s concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo’s derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry.</p>}},
author = {{Deppman, Airton and Megías, Eugenio and Pasechnik, Roman}},
issn = {{1099-4300}},
keywords = {{fractal derivatives; fractional derivatives; fractional differential equations; nonextensive statistics; q-calculus}},
language = {{eng}},
number = {{7}},
publisher = {{MDPI AG}},
series = {{Entropy}},
title = {{Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus}},
url = {{http://dx.doi.org/10.3390/e25071008}},
doi = {{10.3390/e25071008}},
volume = {{25}},
year = {{2023}},
}