Abstract
Until now, classical models of clusters’ fission remain unable to fully explain strange phenomena like the phenomenon of shattering (Ziff and McGrady, 1987) and the sudden appearance of infinitely many particles in some systems having initial finite number of particles. That is why there is a need to extend classical models to models with fractional derivative order and use new and various techniques to analyze them. In this paper, we prove the existence of strongly continuous solution operators for nonlocal fragmentation models with Michaud time derivative of fractional order (Samko et al., 1993). We focus on the case where the splitting rate is dependent on size and position and where new particles generating from fragmentation are distributed in space randomly according to some probability density. In the analysis, we make use of the substochastic semigroup theory, the subordination principle for differential equations of fractional order (Prüss, 1993, Bazhlekova, 2000), the analogy of Hille-Yosida theorem for fractional model (Prüss, 1993), and useful properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005). We are then able to show that the solution operator to the full model is positive and contractive.
Highlights
The dynamics of clusters’ fragmentation occurs in many branches of natural sciences ranging from physics, through chemistry, engineering, biology, ecology, and numerous domains of applied sciences, such as the depolymerization, the rock fractures, and breakage of droplets
There exists a vast literature on classical fragmentation models and many of them have been deeply analyzed in different works where authors investigated their conservativeness, honesty, and existence of solutions
We proved that there is a strongly continuous solution operator, positive and contractive, to nonlocal fragmentation models with Michaud time derivative of fractional order
Summary
The dynamics of clusters’ fragmentation occurs in many branches of natural sciences ranging from physics, through chemistry, engineering, biology, ecology, and numerous domains of applied sciences, such as the depolymerization, the rock fractures, and breakage of droplets. At an infinitesimal and bounded scale, the rate of accumulation or loss of matter in the system is characterized by the classical derivative d/dt This leads us to three essential motivations for considering, in this paper, the fractional nonlocal and randomly position structured fragmentation model. Most of the infinitesimal spaces for complex models can contain hindering or obstacles (of various sizes) where the variable under study is temporarily parked or stuck. In this condition, the classical d/dt will no longer replicate, with certitude, the real picture of accumulation or loss. The fractional version of the same type of model, namely, the fractional, nonlocal, and randomly position structured fragmentation model, is analyzed with full description given
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