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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...

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Detalles Bibliográficos
Autores principales: Deppman, Airton, Megías, Eugenio, Pasechnik, Roman
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2023
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.
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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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