A Mathematical Analysis of Fractional Fragmentation Dynamics with Growth

Abstract

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 rater. 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 when1/r(x)is integrable atx0≥0, the minimum size of a cell. We restrict our analysis to the case of integrability ofr-1atx0. 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.