Coupled approach and its convergence analysis for aggregation and breakage models: Study of extended temporal behaviour

Population balance equations (PBEs) play a significant role in describing the dynamics of star formation in astrophysics, cheese manufacturing in dairy sciences, depolymerization in chemical engineering, and granule preparation in the pharmaceutical industry. However, obtaining analytical solutions for these equations remains a formidable challenge due to complex integral terms. In this work, we propose a generalized approach based on He's variational iteration method which enables the efficient estimation of series solutions for breakage and aggregation PBEs for shorter time domains. To extend the solutions for longer time periods and larger size domains, the series solutions obtained by the coupled method are further expanded using powerful Padé approximants. Moreover, a thorough convergence analysis is conducted in the Banach space. The testing of the new approach is done by considering several gelling and non gelling kernels, and the results are validated against the existing analytical solutions.