Homotopy analysis method and its convergence analysis for a nonlinear simultaneous aggregation-fragmentation model

In this article, we focus on addressing the missing convergence analysis of the homotopy analysis method (HAM) for solving pure aggregation and pure fragmentation population balance equations [Kaur et al. (2023), J. Math. Anal. Appl., 512(2), 126166]. This technique is further extended to determine analytical series solutions for a simultaneous aggregation-fragmentation (SAF) population balance equation. The convergence analysis of the extended approach for a SAF equation is performed using the concept of contraction mapping in the Banach space. The HAM method enables us to derive recursive formulas to obtain series solutions, distinguishing it from traditional numerical approaches. One noteworthy advantage of HAM is its capability to solve both linear and nonlinear differential equations without resorting to discretization, while incorporating a convergence control parameter. Given the complex nature of the SAF equation, only a single analytical solution has been available, specifically for a constant aggregation kernel and a binary breakage kernel with a linear selection function. However, our study presents new series solutions for the number density functions, considering the combination of sum and product aggregation kernels with binary breakage kernels and linear/quadratic selection functions. These particular solutions have not been previously documented in the existing literature. To verify the accuracy and efficiency of the proposed approach, the results with the finite volume scheme [Singh et al. (2021), J. Comput. Phys., 435, 110215] for establishing the accuracy and effectiveness of the proposed approach.