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

Emile F. Doungmo Goufo; Morgan Kamga Pene; Jeanine N. Mwambakana

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Emile F. Doungmo Goufo, Morgan Kamga Pene, and Jeanine N. Mwambakana. "Duplication in a model of rock fracture with fractional derivative without singular kernel." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/275969>.

@article{EmileF2015,
abstract = {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.},
author = {Emile F. Doungmo Goufo, Morgan Kamga Pene, Jeanine N. Mwambakana},
journal = {Open Mathematics},
keywords = {Caputo-Fabrizio fractional derivative; Differentiation without singular kernel; Duplicated poles; Breakup dynamics; Generalized functions; Numerical approximations},
language = {eng},
number = {1},
pages = {null},
title = {Duplication in a model of rock fracture with fractional derivative without singular kernel},
url = {http://eudml.org/doc/275969},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Emile F. Doungmo Goufo
AU - Morgan Kamga Pene
AU - Jeanine N. Mwambakana
TI - Duplication in a model of rock fracture with fractional derivative without singular kernel
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - 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.
LA - eng
KW - Caputo-Fabrizio fractional derivative; Differentiation without singular kernel; Duplicated poles; Breakup dynamics; Generalized functions; Numerical approximations
UR - http://eudml.org/doc/275969
ER -

References

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