Topics in coagulation-fragmentation equations
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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) including gelation. 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).