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, Tome 13 (2015), / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:275969

Bibliographie

[1] Atangana A., Nieto J.J., Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Advances in Mechanical Engineering 2015 (in press)

[2] Atangana A., Doungmo Goufo E.F., A model of the groundwater flowing within a leaky aquifer using the concept of local variable order derivative, Journal of Nonlinear Science and Applications 2015, 8(5), 763–775 | Zbl 1330.76128

[3] Anderson W.J, Continuous-Time Markov Chains, An Applications-Oriented Approach, Springer Verlag, New York 1991 | Zbl 0731.60067

[4] Blatz R., Tobobsky J.N. Note on the kinetics of systems manifesting simultaneous polymerization-depolymerization phenomena. J. Phys. Chem. 1945, 49(2), 77–80; DOI: 10.1021/j150440a004 [Crossref]

[5] Cornelius R.R., Voight B., Real-time seismic amplitude measurement (RSAM) and seismic spectral amplitude measurement (SSAM) analyses with the Materials Failure Forecast Method (FFM), June 1991 explosive eruption at Mount Pinatubo. In: Punongbayan RS, Newhall CG (eds) Fire and mud. Eruptions and lahars of Mount Pinatubo, Philippines. University of Washington Press, Seattle 1996, 249–267

[6] Caputo M., Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Ast. Soc. 1967 13(5), 529–539, reprinted in: Fract. Calc. Appl. Anal. 2008, 11(1), 3–14

[7] Caputo M., Fabrizio M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 2015 1(2), 1–13

[8] Doungmo Goufo E.F., A mathematical analysis of fractional fragmentation dynamics with growth, Journal of Function Spaces 2014, Article ID 201520, 7 pages, http://dx.doi.org/10.1155/2014/201520 [Crossref][WoS] | Zbl 06335886

[9] Doungmo Goufo E.F., A biomathematical view on the fractional dynamics of cellulose degradation, Fractional Calculus and Applied Analysis 2015, 18(3), 554–564 | Zbl 1316.26004

[10] Doungmo Goufo E.F., Mugisha S.B., Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order, Central European Journal of Mathematics 2015, 13(1), 351–362, DOI: 10.1515/math-2015–0033 [Crossref]

[11] Lachowicz M., Wrzosek D., A nonlocal coagulation-fragmentation model. Appl. Math. (Warsaw) 2000 27(1), 45–66 | Zbl 0994.35054

[12] Khan Y., Wu Q., Homotopy Perturbation Transform Method for nonlinear equations using He’s Polynomials, Computers and Mathematics with Applications 2011, 61(8), 1963–1967. | Zbl 1219.65119

[13] Khan Y., Sayevand K., Fardi M., Ghasemi M., A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations, Applied Mathematics and Computation 2014, 249, 229–236. | Zbl 1338.65285

[14] Khan Y., Wu Q., Faraz N., Yildirim A., Madani M., A new fractional analytical approach via a modified Riemann-Liouville derivative, Applied Mathematics Letters October 2012, 25(10), 1340–1346 [WoS][Crossref] | Zbl 1251.65101

[15] Kilburn C.R.J., Multiscale fracturing as a key to forecasting volcanic eruptions. J Volcanol Geotherm Res 2003, 125, 271–289.

[16] Mark H., Simha R., Degradation of long chain molecules, Trans. Faraday 1940, 35, 611–618.

[17] Norris J.R., Markov Chains, Cambridge University Press, Cambridge 1998

[18] Tsao G.T., Structures of Cellulosic Materials and their Hydrolysis by Enzymes, Perspectives in Biotechnology and Applied Microbiology 1986, 205–212

[19] Yang X.J., Baleanu D., Srivastava H.M., Local fractional similarity solution for the diffusion equation defined on Cantor sets, Applied Mathematics Letters 2015, 47, 54–60 | Zbl 06477315

[20] Ziff R.M., McGrady E.D., The kinetics of cluster fragmentation and depolymerization, J. Phys. A 1985, 18, 3027-3037.

[21] Ziff R.M., McGrady E.D., Shattering transition in fragmentation, Phys. Rev. Lett. 1987, 58(9), 892–895 [Crossref]