Sectional (zero order) methods constitute a very important class of methods for the solution of the population balance equation offering distinct advantages compared to their competitors, namely, higher order and moment methods. For the last ten years a particular sectional method, the so called fixed pivot technique [1] has been the most extensively used in the scientific community for the sol...

In this paper we develop an efficient stochastic method to solve the time evolution of a bivariate population balance equation which has been developed for modelling nano-particle dynamics. We have adapted the existing stochastic models used in the study of coagulation dynamics to solve a variant of the sintering-coagulation equation proposed by Xiong & Pratsinis. Hitherto stochastic models bas...

For the standard one sector stochastic optimal growth model, we outline a new set of conditions for a policy function that satisfies the Ramsey-Euler equation to be optimal. An interior Ramsey-Euler policy function is optimal if (and only if) it is continuous or alternatively, if both consumption and investment are non-decreasing in output. One does not need to verify the transversality conditi...

The objective of the present paper is to develop a population balance approach for modelling soot formation that distinguishes between coalescence and aggregation accounts finite-rate fusing primary particles within aggregates, while providing numerically accurate description particle surface growth oxidation aggregates. To this end, recently developed conservative finite volume sectional metho...

This paper reflects the implementation of a reliable technique which is called $left(frac{G'}{G}right)$-expansion  ethod for  constructing exact travelling wave solutions of nonlinear partial  differential equations. The proposed algorithm has been successfully tested on two two selected equations, the balance numbers of which are not positive integers namely Kundu-Eckhaus equation and  Derivat...

In this work, we successfully construct the new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. The idea introduced in this paper can be applied to other nonlinear evolution equations.

Acknowledgements I would like to thank Dr Markus Kraft for his direction and encouragement throughout , Professor James Norris and Ismael Bailleuil from DPMMS for the huge amount of input into this work. Summary In this thesis, two classes of new stochastic algorithms for calculating parametric derivatives of population balance solutions are presented. Currently, it is very difficult to produce...