纯粹的破损过程的群体平衡方程的对称及精确解  

Symmetries and Exact Solutions of Population Balance Equation with Pure Breakage Processes

作　　者：林府标 张千宏 LIN Fubiao;ZHANG Qianhong(School of Mathematics and Statistics,Guizhou University of Finance and Economics,Guiyang 550025,China)

机构地区：[1]贵州财经大学数统学院,贵州贵阳550025

出　　处：《杭州师范大学学报（自然科学版）》2021年第3期295-303,共9页Journal of Hangzhou Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11761018,11361012);贵州省科技计划基金项目(黔科合基础[2019]1051);贵州省科技厅科学技术基金项目([2020]1Y008);贵州省教育厅创新群体项目(黔教合KY字[2021]015);贵州省教育厅青年科技人才成长项目(黔教合KY字[2017]150);贵州财经大学校级科研基金项目(2018XYB04);贵州财经大学创新探索及学术新苗项目(黔科合平台人才[2017]5736-020)。

摘　　要：建立和研究了不含消沉项而仅具有纯粹的破损过程的积分-偏微分方程(群体平衡方程)模型.首先采用伸缩变换群分析方法找到了积分-偏微分方程所接受的伸缩变换群及对称,并将积分-偏微分方程转化为纯偏微分方程,利用经典的李群分析方法获得了纯偏微分方程所接受的李群及对称.然后综合伸缩变换群和经典的李群分析方法获得的结果,找到了原积分-偏微分方程所接受的对称、群不变解和约化的积分-常微分方程.最后利用观察试凑函数法发现了约化的积分-常微分方程的精确解,进而给出了原积分-偏微分方程的显式精确解和种群粒子的累积质量及密度分布函数.An integro-partial differential equation(population balance equation) with no depression term and only pure breakage processes was built and investigated. Firstly, the admitted scaling transformation groups and symmetry of the integro-partial differential equation were found by using the method of scaling transformation group analysis. In addition, the integro-partial differential equation was transformed to a pure partial differential equation, the admitted Lie groups and symmetry of the pure partial differential equation were obtained by using the method of classical Lie group analysis. Secondly, all admitted symmetries, group invariant solutions and reduced equations of the integro-partial differential equation were presented by scaling transformation analysis and classical Lie group analysis. Finally, all exact solutions for the reduced integro-ordinary differential equation were found by using the methods of observation trial functions. Hence the explicit exact solutions for the original integro-partial differential equation was provided. The cumulative mass and density distribution functions of population particles were also given.

关 键 词：群体平衡方程 纯粹的破损过程 李群分析 对称 精确解 

分 类 号：O175.5[理学—数学] O175.6[理学—基础数学]