The population balance equation (PBE) is one of the most popular integro-differential equations modeled for several industrial processes. The solution to this equation is usually solved using a numerical approach as the analytical solutions of such equations are not obtained easily. Typically, the available analytical solutions are limited and are based on momentous Laplace transform. In this study, the reduced equations of the PBE are obtained via the group analysis method. Two particulate cases involving aggregation, growth and nucleation are selected, the determining equations are solved and the reduced equations are solved via approximate methods. The approximate method involves the target solution of the nonlinear evolution equation, here the PBE, to be expressed as a polynomial in an elementary function which satisfies a particular ordinary differential equation termed as an auxiliary equation.

中文翻译：

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通过对称性研究涉及聚合和增长项的人口平衡方程

人口平衡方程 (PBE) 是为多个工业过程建模的最流行的积分微分方程之一。该方程的解通常使用数值方法求解，因为此类方程的解析解不容易获得。通常，可用的解析解是有限的，并且基于重要的拉普拉斯变换。在本研究中，PBE 的简化方程是通过组分析方法获得的。选取涉及聚集、生长和成核的两种粒子情况，求解确定方程，并通过近似方法求解简化方程。近似方法涉及非线性演化方程的目标解，这里是 PBE，