一类群体平衡方程的尺度变换群分析及显式精确解

研究了一类既存在增长过程又存在破损过程的群体平衡方程的精确解法。用尺度变换群分析法得到群体平衡方程的部分对称、群不变解和约化积分-常微分方程。用试探函数法探求约化积分-常微分方程，得到群体平衡方程的显式精确解，并分析了该显式精确解的动力学特性。所得群不变解能解释实体模型，显式精确解可检验数值解的正确性和精确度。

Analysis of scaling transformation group and explicit exact solutions of the population balance equation involving breakage and growth processes

The analytical method and exact solutions of the population balance equation involving growth and breakage processes are investigated in this paper. Partial symmetries, group invariant solutions and reduced integro-ordinary differential equations of the population balance equation are found by scaling transformation group analysis. Explicit exact solutions of the population balance equation are obtained by solving the reduced integro-ordinary differential equations with the method of trial function. The dynamic behavior of explicit exact solutions is analyzed. The obtained group invariant solutions can provide interpretation for the physical processes model, on the other hand, these obtained exact solutions can also be used to verify the correctness and accuracy of numerical solutions.