Mathematical and numerical analysis of the discrete fragmentation coagulation equation with growth, decay and sedimentation.
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Date
2018
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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.
Description
Doctoral degree. University of KwaZulu-Natal, Durban.