用伸缩变换群分析法探究群体平衡方程的自相似解

Self-similar Solutions of the Population Balance Equation,Using Scaling Group Analysis

改进了的李群分析方法用于积分-偏微分方程(群体平衡方程)十分复杂,问题的本质在于求解积分-偏微分方程的决定方程既棘手又困难,探究决定方程的方法依赖于原积分-偏微分方程本身的结构特征和性质.相反,采用伸缩变换群分析方法探索积分-偏微分方程的自相似解既简单又方便.论文利用伸缩变换群分析方法研究了积分-偏微分方程,获得了积分-偏微分方程的显式真实解、自相似解和约化的积分-常微分方程.

An application of the Lie group analysis method as developed for integrodifferential equations(population balance equation)is quite complicated,due to the fact that solving the determining equations of integro-differential equations is typically difficult.Even more,the ways of solving determining equations depend on the original integrodifferential equations studied.Conversely,for simplicity and convenience,the study of new integro-differential equations is started by considering the self-similar solutions using a scaling group.Population balance equations considered in the present paper involve both a partial differential equation and integro-differential equation.All explicit physical solutions and self-similar solutions were presented by using the scaling group analysis method.Analysis of the reduced equations was also provided.