The Smoluchowski's aggregation equation has applications in the field of bio-pharmaceuticals \cite{zidar2018characterisation}, financial sector \cite{PUSHKIN2004571}, aerosol science \cite{shen2020efficient} and many others. Several analytical, numerical and semi-analytical approaches have been devised to calculate the solutions of this equation. Semi-analytical methods are commonly employed since they do not require discretization of the space variable. The article deals with the introduction of a novel semi-analytical technique called the optimized decomposition method (ODM) (see \cite{odibat2020optimized}) to compute solutions of this relevant integro-partial differential equation. The series solution computed using ODM is shown to converge to the exact solution. The theoretical results are validated using numerical examples for scientifically relevant aggregation kernels for which the exact solutions are available. Additionally, the ODM approximated results are compared with the solutions obtained using the Adomian decomposition method (ADM) in \cite{singh2015adomian}. The novel method is shown to be superior to ADM for the examples considered and thus establishes as an improved and efficient method for solving the Smoluchowski's equation.

翻译：Smoluchowski的聚合方程式在生物制药学\cite{zidar2018字符化}、金融部门\cite{PUSHKIN2004571}、气溶胶科学\cite{shen2020效能}和其他许多领域都有应用。已经设计了几种分析、数字和半分析方法来计算这个方程式的解决方案。半分析方法通常使用,因为它们不需要对空间变量进行离散。文章涉及采用一种叫作优化分解法(ODM)的新型半分析技术(见\cite{odibat2020Opptimization})来编解析与此相关的全方程式差异方程式的解决方案。使用 ODM 计算的系列解决方案与确切解决方案一致。理论结果通过科学相关总库的数值示例得到验证。此外, ODM 比较的结果与使用Adomian解剖法获得的解决方案(ODM) (ADMDM) 和SDMA-decompliadroup 方法(ADM) 被考虑后,SDM 方法将SDM 和SDVADMDUDUDD=S的解决方案加以验证。