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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 13, Issue 1


Volume 13 (2015)

Duplication in a model of rock fracture with fractional derivative without singular kernel

Emile F. Doungmo Goufo / Morgan Kamga Pene / Jeanine N. Mwambakana
  • of Science, Mathematics and Technology Education, University of Pretoria, Pretoria, South Africa
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-11-27 | DOI: https://doi.org/10.1515/math-2015-0078


We provide a mathematical analysis of a break-up model with the newly developed Caputo-Fabrizio fractional order derivative with no singular kernel, modeling rock fracture in the ecosystem. Recall that rock fractures play an important role in ecological and geological events, such as groundwater contamination, earthquakes and volcanic eruptions. Hence, in the theory of rock division, especially in eco-geology, open problems like phenomenon of shattering, which remains partially unexplained by classical models of clusters’ fragmentation, is believed to be associated with an infinite cascade of breakup events creating a ‘dust’ of stone particles of zero size which, however, carry non-zero mass. In the analysis, we consider the case where the break-up rate depends of the size of the rock breaking up. Both exact solutions and numerical simulations are provided. They clearly prove that, even with this latest derivative with fractional order and no singular kernel, the system describing crushing and grinding of rocks contains (partially) duplicated fractional poles. According to previous investigations, this is an expected result that provides the new Caputo-Fabrizio derivative with a precious and promising recognition.

Keywords: Caputo-Fabrizio fractional derivative; Differentiation without singular kernel; Duplicated poles; Breakup dynamics; Generalized functions; Numerical approximations


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About the article

Received: 2015-05-31

Accepted: 2015-08-08

Published Online: 2015-11-27

Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0078.

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©2015 Doungmo Goufo et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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