Analytical and numerical treatments of a coagulation population balance model

The Redner–Ben-Avraham–Kahng (RBK) coagulation model, initially proposed as a discrete framework for investigating cluster growth kinetics, has recently been reformulated to encompass a continuous representation. While the existence, uniqueness, and long-term dynamics of solutions for the continuous model have been examined, both analytical and numerical solutions have yet to be thoroughly addressed. This study undertakes a comprehensive investigation of the continuous RBK coagulation model utilizing both numerical and semi-analytical methodologies, specifically the Finite Volume Method (FVM) and the Homotopy Perturbation Method (HPM). Analytical expressions for the number density function are derived for a variety of coagulation kernels, including constant, sum, product, and bilinear kernels, based on exponential and gamma initial distributions. The efficacy of the HPM is rigorously assessed through an extensive convergence analysis, which encompasses the order of convergence and error estimates pertinent to the series solutions. Furthermore, the outcomes obtained from HPM are validated against those derived from the established FVM, thereby demonstrating the reliability and effectiveness of HPM in addressing the continuous RBK model.