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Open AccessArticle

Diffusion on Middle-ξ Cantor Sets

Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran
Department of Mathematics, Cankaya University, Ankara 06530, Turkey
Institute of Space Sciences, P.O. Box, MG-23, R 76900 Magurele-Bucharest, Romania
Author to whom correspondence should be addressed.
Entropy 2018, 20(7), 504;
Received: 23 May 2018 / Revised: 6 June 2018 / Accepted: 6 June 2018 / Published: 2 July 2018
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
In this paper, we study Cζ-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the Cζ-calculus on the generalized Cantor sets known as middle-ξ Cantor sets. We have suggested a calculus on the middle-ξ Cantor sets for different values of ξ with 0<ξ<1. Differential equations on the middle-ξ Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given. View Full-Text
Keywords: Hausdorff dimension; middle-ξ Cantor sets; staircase function; Cζ-calculus; diffusion on fractal; random walk Hausdorff dimension; middle-ξ Cantor sets; staircase function; Cζ-calculus; diffusion on fractal; random walk
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Khalili Golmankhaneh, A.; Fernandez, A.; Khalili Golmankhaneh, A.; Baleanu, D. Diffusion on Middle-ξ Cantor Sets. Entropy 2018, 20, 504.

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