Global Solvability of a Continuous Model for Nonlocal Fragmentation Dynamics in a Moving Medium

Noutchie, Suares Clovis Oukouomi; Goufo, Emile Franc Doungmo

doi:10.1155/2013/320750

Cited by 12 publications

(19 citation statements)

References 16 publications

(26 reference statements)

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“…Because the uniqueness of solutions to the systems of type (34)- (35) is proved to be a more difficult problem [12,23], we restrict our analysis to a smaller class of functions, so we introduce the following class of Banach spaces (of distributions with finite higher moments):…”

Section: Well-posedness For the Break-up Part Of The Modelmentioning

confidence: 99%

Evolution equations with a parameter and application to transport-convection differential equations

Goufo¹

2017

Turk J Math

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We deeply investigate the well-posedness of models taking the formt is a derivative with the fractional parameter β and A is a closed densely defined operator in a Banach space. We show that, unlike other systems, solutions of our models are not governed by Mittag-Leffler functions and their variants. We extend and adapt Peano's idea to our models and establish conditions for existence and uniqueness of solutions. In particular, relations between the two-parameter solution operator, its resolvent, and its generator are provided; the issue of subordination and prolongation principles are addressed; and a way to approximate the generalized solution is presented. Finally, application to transport-convection differential equations is performed in the space of distributions with finite higher moments to show how their well-posedness can be addressed.

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Section: Well-posedness For the Break-up Part Of The Modelmentioning

confidence: 99%

Evolution equations with a parameter and application to transport-convection differential equations

Goufo¹

2017

Turk J Math

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“…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%

“…is a well-defined positive operator [41]. Then (4)- (5) can be written as an abstract Cauchy problem in X 1 :…”

Section: Fragmentation Differential Equationmentioning

confidence: 99%

“…It is well known [6,19,41] that an operator e Q 2 G.M; !/ means e Q generates a C 0 -semigroup .G e Q .t // t>0 so that there exists M > 0 and ! such that…”

Section: Fractional Fission Evolution Equationmentioning

confidence: 99%

“…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. 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%

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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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Approximation result for non-autonomous and non-local rock fracture models

Goufo

Kubeka

2018

Japan J. Indust. Appl. Math.

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Global Solvability of a Continuous Model for Nonlocal Fragmentation Dynamics in a Moving Medium

Cited by 12 publications

References 16 publications

Evolution equations with a parameter and application to transport-convection differential equations

Evolution equations with a parameter and application to transport-convection differential equations

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

Approximation result for non-autonomous and non-local rock fracture models

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