Exact Solutions of Fragmentation Equations with General Fragmentation Rates and Separable Particles Distribution Kernels

Noutchie, Suares Clovis Oukouomi; Goufo, Emile Franc Doungmo

doi:10.1155/2014/789769

Cited by 7 publications

(5 citation statements)

References 16 publications

(16 reference statements)

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“…Note that the analytical solution (4) in the Laplace space includes the previously studied case of A = 1 and v 0 → ∞. [22] The analytical solution in real space (v, t) can be obtained by inverting expression (4). Now taking into account expressions (3), the inverse Laplace transforms [24] 1 p + λ → exp(−λt) ,…”

Section: Model and Its Exact Solutionmentioning

confidence: 99%

“…Note that expression (5) transforms to the previously known distribution function [22] in the limiting case of infinitely large initial particle volume (v 0 → ∞) and A = 1. It is easily seen that the distribution function increases and narrows with increasing the process time (Fig.…”

Section: Model and Its Exact Solutionmentioning

confidence: 99%

“…All pre-viously obtained unsteady-state analytical distributions have been derived with allowance for the hypothesis of infinite volume (see, among others, Refs. [18][19][20][21][22]). We demonstrate below that the presence of a finite initial particle volume substantially changes the resultant particlesize distribution function.…”

Section: Introductionmentioning

confidence: 99%

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On the Theory of Fragmentation Process with Initial Particle Volume

Alexandrov¹

2017

Commun. Theor. Phys.

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Section: Model and Its Exact Solutionmentioning

confidence: 99%

Section: Model and Its Exact Solutionmentioning

confidence: 99%

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On the Theory of Fragmentation Process with Initial Particle Volume

Alexandrov¹

2017

Commun. Theor. Phys.

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“…In this way we compute few terms to approximate the solution of the fractional fragmentation Cauchy problem (28)- (29) and the asymptotic solution is given by (40). The existence of solutions to (28)- (29) is guaranteed by the condition (30).…”

Section: Homotopy Perturbation For the Loss Part Of Fractional Fissiomentioning

confidence: 99%

“…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. In [40], the authors provided explicit solutions to clusters' fragmentation equations with general fragmentation rates, giving a general framework for understanding particles distributions in fragmentation processes as time evolves. Although the result is a breakthrough in the analysis of clusters's fragmentation equations with arbitrary fragmentation rates, the phenomenon of shattering remains partially unexplained.…”

Section: Model's Motivation and Introductionmentioning

confidence: 99%

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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Tuberculosis model with relapse via fractional conformable derivative with power law

Khan

Gómez-Aguilar

2019

Math Methods in App Sciences

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Funding information Cátedras CONACyT para jóvenes investigadores 2014; SNI-CONACyTThe present study explores the tuberculosis dynamics with relapse with a nonlocal conformable derivative in the Caputo sense. The real data of tuberculosis cases since 2002 to 2017 are used to the set the parameters. The numerical results with the realistic parameters are chosen and present the graphical results. Further, we assign different values to the fractional parameters and and discuss its effect on the system variables. The use of these two fractional operators on the model simultaneously with realistic data gives reliable results. KEYWORDSCaputo operator, mathematical model, nonlocal conformable derivative, numerical results, tuberculosis relapse model MSC CLASSIFICATION 26A33; 92D30; 34Dxx Math Meth Appl Sci. 2019;42:7113-7125. wileyonlinelibrary.com/journal/mma

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Exact Solutions of Fragmentation Equations with General Fragmentation Rates and Separable Particles Distribution Kernels

Cited by 7 publications

References 16 publications

On the Theory of Fragmentation Process with Initial Particle Volume

On the Theory of Fragmentation Process with Initial Particle Volume

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

Tuberculosis model with relapse via fractional conformable derivative with power law

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