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

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.