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Journal of Function Spaces
Volume 2014, Article ID 201520, 7 pages
Research Article

A Mathematical Analysis of Fractional Fragmentation Dynamics with Growth

Department of Mathematical Sciences, University of South Africa, Florida Science Campus, Gauteng 0003, South Africa

Received 21 April 2014; Accepted 25 July 2014; Published 7 August 2014

Academic Editor: Gestur Ólafsson

Copyright © 2014 Emile Franc Doungmo Goufo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We make use of the theory of strongly continuous solution operators for fractional models together with the subordination principle for fractional evolution equations (Bazhlekova (2000) and Prüss (1993)) to analyze and show existence results for a fractional fragmentation model with growth characterized by its growth rate . Indeed, strange phenomena like the phenomenon of shattering (McGrady and Ziff (1987)) and the sudden appearance of infinite number of particles in some systems with initial finite particles number could not be fully explained by classical models of fragmentation or aggregation. Then, there is an increasing volition to try new approaches and extend classical models to fractional ones. In the growth model, one of the major challenges in the analysis occurs when is integrable at , the minimum size of a cell. We restrict our analysis to the case of integrability of at . This case needs more considerations on the boundary condition, which, in this paper, is the McKendrick-von Foerster renewal condition. In the process, some properties of Mittag-Leffler relaxation function Berberan-Santos (2005) are exploited to finally prove that there is a positive solution operator to the full model.