|dc.description.abstract||In this thesis we study the mathematics of a model for the dynamics of cluster growth.
The sizes of the clusters change in time as the clusters undergo coagulation and
fragmentation events. The equations are for j = 1,2,...
c′ j =1 2j−1 X k=1[aj−k,kcj−kck −bj−k,kcj]−∞ X k=1[aj,kcjck −bj,kcj+k] (1.1)
where cj(t) is the concentration of clusters of size j and aj,k,bj,k are the constant rates
of coagulation and fragmentation.
Chapter 1 reviews some results on (1.1) and introduces some mathematical tools
used in the thesis. It also introduces the concept of gelation, which is the formation
of an inﬁnite cluster leading to the loss of mass conservation.
In Chapter 2 we study gelation in (1.1) and discuss ﬁnite dimensional approxima
tions which are used for numerical studies. We explain why a certain ﬁnite dimen
sional system which does not conserve density is suitable for numerical studies of (1.1)
All solutions of the ﬁnite dimensional system converge to zero and Chapter 3
deals with the asymptotic behaviour. For the case in which the coagulation and
fragmentation terms are non zero and satisfy a detailed balance condition, we obtain
a general result on the asymptotic decay. However, for the pure coagulation case
(bj,k = 0), we show that a wide variety of asymptotics is possible.
Chapter 4 is concerned with a model for the treatment of Alzheimer’s disease.
The model is a modiﬁed form of (1.1). We prove some mathematical results for the
system and obtain an approximate formula for the decay rate.
Chapter 5 deals with numerical approximations to the continuous version of (1.1).
We consider a piecewise constant in space approximation in both collocation and the
Galerkin formulation. Numerical results indicate that the Galerkin ﬁnite element
method has second order accuracy. These approximations of the continuous problem
are themselves discrete systems like (1.1).||en