Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates

Goufo, Emile Franc Doungmo; Noutchie, Suares Clovis Oukouomi

doi:10.1016/j.crma.2013.09.023

Cited by 10 publications

(16 citation statements)

References 10 publications

(18 reference statements)

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“…(a): By condition (6), the operator A is bounded linear operator. Then, we know that [16,20,41] it is the infinitesimal generator of a positive semigroup of contractions (or substochastics semigroup defined in Definition 2.1) and the assertion follows.…”

Section: Fractional Fission Evolution Equationmentioning

confidence: 94%

“…For more details about analysis on fragmentation differential equations, we refer the reader to [6,16,20,24,34,41,50,54] and the references therein. The classic model for fragmentation process is given by the integrodifferential equation…”

Section: Fragmentation Differential Equationmentioning

confidence: 99%

“…Conservative and nonconservative regimes for fragmentation equations have been thoroughly investigated -see [20,55], and, in particular, the breach of the mass conservation law (called shattering) has been attributed to a phase transition creating a dust of "zerosize" particles with nonzero mass, which are beyond the model's resolution. Shattering can be interpreted from the probabilistic point of view as the explosion in the Markov process describing fragmentation [50], and from an analytic point of view as dishonesty of the semigroup associated with the model.…”

Section: Model's Motivation and Introductionmentioning

confidence: 99%

“…Transport-type models With convection were investigated in [41] where the author showed that the convection part does not affect the breach of the conservation laws. In [20] the authors studied the non local fragmentation and showed that the process is conservative if at infinity daughter particles tend to go back into the system with a high known probability.…”

Section: Model's Motivation and Introductionmentioning

confidence: 99%

“…These methods include semigroup theory [6,21,43], perturbation theory [2,3,20], approximation techniques [19,33], and probabilistic methods [23]. The efficiency of these methods is limited as these problems are reformulated in abstract spaces that are norm dependent and the overall behavior of the dynamics changes radically as different metrics are included in the system.…”

Section: Model's Motivation and Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Goufo¹,

Mugisha²

2015

Open Mathematics

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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

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Section: Fractional Fission Evolution Equationmentioning

confidence: 94%