This thesis is devoted to the mathematical and numerical analysis for the continuous coagulation-fragmentation equation. This is a partial integro-differential equation. There have been several investigations of existence and uniqueness of solutions to the coagulation and binary fragmentation equation with different classes of kernels. However, the case of multiple fragmentation was almost ignored. The first aim of this work is to prove the existence of solutions to the continuous coagulation and multiple fragmentation equation for large classes of kernels. Here we would like to cover those coagulation kernels which are not included in the previous literature for the study of the continuous coagulation equation with multiple fragmentation. It is also of great interest to investigate the uniqueness of solutions. However, in order to prove the uniqueness, we need more restrictive conditions on the kernels. The second aim is to demonstrate the uniqueness of mass conserving solutions to the continuous coagulation and binary fragmentation equation. In this case, the existence of mass conserving solutions was established in Escobedo et al.\ \cite{Escobedo:2003} for a large class of coagulation kernels with strong fragmentation. This strong fragmentation prevents the occurrence of the gelation phenomenon and gives the existence of mass conserving solutions when the class of coagulation kernels grows beyond linearity. Note that the gelation phenomenon usually leads to solutions which are not mass conserving. Therefore, the proof of uniqueness requires additional growth conditions on the fragmentation kernels. The third target is to extend the previous existence result for the coagulation and multiple fragmentation equation. In this work we wish to include some classical multiple fragmentation kernels which %are singular at zero. These types of fragmentation kernels are not covered in the existence result mentioned above. It should also be remarked that the classes of coagulation kernels are identical to those in the above result. The next goal is to develop the convergence analysis of sectional methods for solving the non-linear pure coagulation equation. Here we examine the most popular of all sectional methods the fixed pivot technique. We investigate the convergence of the fixed pivot scheme on five different grid types. We found that the scheme is second order accurate on uniform and non-uniform smooth grids while it shows first order accuracy on locally uniform grids. The undesirable result is that the scheme is not convergent on oscillatory and random grids. Finally, we demonstrate practical significance of the mathematical results by performing a few numerical simulations. The fixed pivot technique gives a consistent over prediction of the solution for the large size particles when applied on coarse grids. To overcome this problem, the cell average technique was introduced which preserves all advantages of the fixed pivot technique and improves the numerical results. Further, we are also interested to evaluate the order of convergence of the cell average technique for the pure coagulation equation by performing several numerical experiments. Then we compare the numerical results with the result obtained by the fixed pivot technique. This cell average technique yields second order accuracy on uniform, non-uniform smooth and locally uniform grids. The scheme turns into a first order accurate method on oscillatory and random grids. Therefore, the cell average technique experimentally shows one order higher accuracy than the fixed pivot technique for locally uniform, oscillatory and non-uniform random grids. The mathematical proof of this higher order remains an open problem.
