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.

Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer

“…However, there were some serious issues that were pointed out against this new finding. First the kernel used was not non-local; second the anti-derivative associated is just the average of the function and its integral [1,2,4,[8][9][10][11]. Therefore it was concluded that, the operator was rather a filter than a fractional derivative.…”

“…Therefore it was concluded that, the operator was rather a filter than a fractional derivative. However we should point out that many works were done around this new finding with great success [1,2,4,10,11]. This derivative was used in mechanical engineering, in groundwater studies, in thermal science, in diffusion model and other [1,2,4,10].…”

Chaos on the Vallis Model for El Niño with Fractional Operators

“…We used the local derivative in one model, the Caputo derivative with power function kernel with singularity and the Caputo-Fabrizio derivative with exponential kernel, which has no singularity [21][22][23][24]. We presented the existence and uniqueness of the model with local derivative and then we derived approximate solutions via iterative method.…”

On numerical solutions for the Caputo-Fabrizio fractional heat-like equation

“…Jafari et al [24] applied the HAM in order to solve the high orderly fractional differential equation analyzed by Diethelmand and Ford [25]. In [26], is produced a mathematical analysis of an example studied the Caputo-Fabrizio fractional derivative, where analytical and calculation advances are fined. Morales-Delgado et al [27] presented LHAM to supply new solutions in case of every two fractional operators.…”