Mathematical and numerical analysis of the discrete fragmentation coagulation equation with growth, decay and sedimentation.

Abstract

Fragmentation-coagulation equations arise naturally in many branches of engineering and science, the applications stretching from astrophysics, blood clotting, colloidal chemistry and polymer science to molecular beam epitaxy. In realistic application, the fragmentation and coagulation are often coupled with growth, decay and/or sedimentation processes. The resulting models are used to describe the evolution of a population in which individuals can grow, coalesce, split or divide, and die. For example, in the phytoplankton dynamics, in addition to forming or breaking of clusters, individuals within them are born or die and so the latter processes must be adequately represented in the models. In the continuous case, the birth or death processes are incorporated into the model by adding an appropriate first order transport term, analogously to the age and size structured McKendrick models. In the discrete case, these vital processes are modelled by adding weighted differences operators. In this study, we focus on the discrete fragmentation-coagulation models with growth, decay or/and sedimentation. The problem is treated as an infinite-dimensional differential equation, which consists of a linear part (fragmentation, growth, decay and sedimentation term) and a nonlinear part (coagulation term), posed in a suitable Banach space, X. We use the theory of semigroups of linear operators, perturbation of positive semigroups and semilinear operators for the mathematical analysis of these models. The linear part of the models is shown to generate a semigroup which is analytic, compact and irreducible and thus has the asynchronous exponential growth property. These results are used to demonstrate the existence of global classical solutions to the semilinear fragmentation-coagulation equations with growth, decay and sedimentation for a class of unbounded coagulation kernels. Theoretical conclusions are supported by numerical simulations.