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Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus
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 fr...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
MDPI
2023
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10378034/ https://www.ncbi.nlm.nih.gov/pubmed/37509954 http://dx.doi.org/10.3390/e25071008 |
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author | Deppman, Airton Megías, Eugenio Pasechnik, Roman |
author_facet | Deppman, Airton Megías, Eugenio Pasechnik, Roman |
author_sort | Deppman, Airton |
collection | PubMed |
description | 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. |
format | Online Article Text |
id | pubmed-10378034 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2023 |
publisher | MDPI |
record_format | MEDLINE/PubMed |
spelling | pubmed-103780342023-07-29 Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus Deppman, Airton Megías, Eugenio Pasechnik, Roman Entropy (Basel) Article 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. MDPI 2023-06-30 /pmc/articles/PMC10378034/ /pubmed/37509954 http://dx.doi.org/10.3390/e25071008 Text en © 2023 by the authors. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). |
spellingShingle | Article Deppman, Airton Megías, Eugenio Pasechnik, Roman Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus |
title | Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus |
title_full | Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus |
title_fullStr | Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus |
title_full_unstemmed | Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus |
title_short | Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus |
title_sort | fractal derivatives, fractional derivatives and q-deformed calculus |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10378034/ https://www.ncbi.nlm.nih.gov/pubmed/37509954 http://dx.doi.org/10.3390/e25071008 |
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