Analytical study for Redner–Ben-Avraham–Kahng coagulating cluster dynamic model

Population balance equations are extensively used to examine the development of aerosols, turbulent gas–liquid flows, colloids, liquid–liquid dispersions, raindrop fragmentation, and pharmaceutical granulation. Due to the complex and nonlinear characteristics of the equation, obtaining an analytical solution for empirical kernels is particularly challenging. To overcome these challenges, in this study, the continuous Redner–Ben-Avraham–Kahng coagulation model is investigated through semi-analytical approaches, specifically employing the homotopy analysis method and the accelerated homotopy analysis method. In the absence of readily available analytical solutions, the approximated series solutions are validated by comparing them with the well-established finite volume method solutions. The flexibility and precision of proposed methods in solving the Redner–Ben-Avraham–Kahng coagulating cluster dynamic model are thoroughly examined. Accelerated homotopy analysis method demonstrates a significant improvement in solution quality over extended periods while requiring fewer terms in the truncated series as compared to homotopy analysis method. Additionally, a detailed convergence analysis of accelerated homotopy analysis method is provided using Banach fixed point theory.