Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order

Abstract

Classical models of clusters' fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order (with 0 < Ä 1). In this article, we introduce and study a model that can be understood as the fractional generalization of the clusters' fission process. We make use of the theory of strongly continuous solution operators for fractional models (analogues of C 0 -semigroups for classical models) and the subordination principle for fractional evolution equations (Bazhlekova, 2000, Prüss, 1993 to analyze and show existence results for clusters' splitting model with derivative of fractional order. In the process, we exploit some properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005), the He's homotopy perturbation (He, 1999) and Kato's type perturbation (Banasiak, 2006) methods. The Cauchy problem for multiplication operator in the fractional dynamics is first considered, before we perturb it. Some additional concepts like Laplace transform, Hille-Yosida theorem and the dominated convergence theorem are use to finally show that there is a solution operator to the full fractional model that is positive and contractive. Model's motivation and introductionThe concepts of fractional derivatives and fractional integral started in 1695 when L'Hospital questioned about the meaning of the operator d n y=dx n if n D 1=2I that is "what if n is fractional?". Leibniz then replied as "d 1=2 x=dx 1=2 will be equal to x p dx W x". Despite its three centuries of age, fractional calculus remains lightly unpopular amongst science and engineering community. However, there is a growing interest in extending the normal calculus with interger orders to noninteger orders (real or complex order) [13,36,39,44] because its applications have attracted a great range of attention in the past few years. Indeed, it has turned out recently that many phenomena in different fields, including sciences, engineering and technology can be described very successfully by the models using fractional order differential equations. As example, the concept of fractional Laplacian operator in the theory of Lévy flights [12] is a typical application of fractional derivatives, which leads to the theory of sub-and superdiffusion, well applicable in reaction-diffusion systems. In the field of mathematical epidemiology, especially the

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

“…Hence, the integral in (10) exists in any interval OE ; 1/ and is absolutely continuous at each x > 0: Thus, differentiating (10) gives the partial differential equation…”