On The Semi-Analytical Solution of Integro-Partial Differential Equations

The breakage and aggregation processes in batch systems had attained highly interest in applied mathematics and engineering fields. In this work, we developed analytical solutions of the particle breakage and aggregation using the population balance equation (PBE) in batch flow systems. To allow explicit solutions, we approximated the particle breakage and aggregation mechanisms by assuming functional forms for breakage and aggregation kernels. In this framework, the Adomian decomposition method (ADM), variational iteration method (VIM) and homotopy perturbation method (HPM) were used to solve the population balance model. These semi-analytical methods overcome the crucial difficulties of numerical discretization and stability that often characterize previous solutions of the PBEs. The obtained results in all cases showed that the predicted particle size distributions converge exactly in a continuous form to that of the analytical solutions using the three methods.