Mixed discrete-continuous fragmentation equations

The work contained in this thesis concerns the development, and the mathematical and numerical analysis, of a new class of hybrid discrete-continuous fragmentation model. The framework is introduced as a potential answer to the occurrence of 'shattering' mass loss, commonly observed in purely continuous fragmentation models. Initially, the study begins by introducing the model, which takes the form of an integro-differential equation, coupled with a system of ordinary differential equations. Once the model has been established, it is subjected to a rigorous mathematical analysis, using the theory and methods of operator semigroups and their generators. Most notably, by applying the theory relating to the Kato-Voigt perturbation theorem, honest substochastic semigroups and operator matrices, the existence of a unique, differentiable solution to the model is established. This solution is also shown to preserve non-negativity and conserve mass. Having determined the existence of a solution, the work continues with the development of a numerical scheme for the approximate solution of the modelling equations. Considering a truncated version of the equations, rewritten in an alternative conservative form, the scheme is built around a finite volume discretisation. Using a standard weak compactness argument, the approximations generated by the numerical scheme are shown to converge (weakly) to a weak solution of the truncated equations. By relating this weak solution to the strong solutions provided by the earlier semigroup analysis, the weak solution is found to be unique and as a consequence, differentiable, non-negative and mass-conserving. The theoretical study is completed with an examination of the effect of varying the truncation point. In particular, establishing that as the length of the truncated interval is increased, in the limit, the original solution to the full model is obtained. Finally, the thesis is completed with a numerical investigation, seeking to experimentally confirm the assertions of the earlier theoretical work and assess the performance of the numerical scheme for a suite of test models.