Population balance models often integrate fundamental kernels, including sum, gelling and Brownian aggregation kernels. These kernels have demonstrated extensive utility across various disciplines such as aerosol physics, chemical engineering, astrophysics, pharmaceutical sciences and mathematical biology for the purpose of elucidating particle dynamics. The objective of this study is to refine the semi-analytical solutions derived from current methodologies in addressing the nonlinear aggregation and coupled aggregation-breakage population balance equation. This work presents a unique semi-analytical approach based on the homotopy analysis method (HAM) to solve pure aggregation and couple aggregation-fragmentation population balance equations, which is an integro-partial differentia equation. By decomposing the non-linear operator, we investigate how to utilize the convergence control parameter to expedite the convergence of the HAM solution towards its precise values in the proposed method.

翻译：群体平衡模型通常整合基本核函数，包括求和、胶凝和布朗聚集核。这些核函数已在气溶胶物理、化学工程、天体物理、制药科学和数学生物学等多个学科中展现出广泛用途，旨在阐明粒子动力学特征。本研究的目标是优化当前方法在求解非线性聚集及耦合聚集-破碎群体平衡方程时提出的半解析解。本文提出了一种基于同伦分析方法（HAM）的独特半解析途径，用于求解纯聚集与耦合聚集-破碎群体平衡方程（一种积分-偏微分方程）。通过分解非线性算子，我们探讨了如何利用收敛控制参数加速HAM解向精确值收敛的过程。