一类群体平衡方程的尺度变换群分析及显式精确解

doi:10.3785/j.issn.1008-9497.2022.01.005

浙江大学学报（理学版）

2022, Vol. 49

Issue (1): 36-40 DOI: 10.3785/j.issn.1008-9497.2022.01.005

数学与计算机科学

一类群体平衡方程的尺度变换群分析及显式精确解

林府标,张千宏(

)

贵州财经大学数统学院，贵州贵阳 550025

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

Fubiao LIN,Qianhong ZHANG(

)

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China

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摘要：

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。

关键词： 积分-偏微分方程; 群体平衡方程; 尺度变换群; 显式精确解

Abstract:

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.

Key words: integro-partial differential equation population balance equation scaling transformation group explicit exact solution

收稿日期: 2019-05-06 出版日期: 2022-01-18

CLC:

O 175.6

基金资助: 国家自然科学基金资助项目(11761018);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学［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)

通讯作者: 张千宏 E-mail: zqianhong68@163.com

作者简介: 林府标（1978—），ORCID：https：//orcid.org/0000-0003-4529-4930，男，博士，副教授，主要从事偏微分方程研究.

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引用本文:

林府标, 张千宏. 一类群体平衡方程的尺度变换群分析及显式精确解[J]. 浙江大学学报（理学版）, 2022, 49(1): 36-40.

Fubiao LIN, Qianhong ZHANG. Analysis of scaling transformation group and explicit exact solutions of the population balance equation involving breakage and growth processes. Journal of Zhejiang University (Science Edition), 2022, 49(1): 36-40.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2022.01.005 或 https://www.zjujournals.com/sci/CN/Y2022/V49/I1/36

算子	$X$	$Y$
$X$	0	$(1 - p) X$
$Y$	$(p - 1) X$	0
$Z$	0	0

表1 子李代数L3的换位子运算结果

序号	生成元	序号	生成元
1	$X, Y, Z$	5	$Y + α Z$
2	$Y, Z$	6	$X + Z$
3	$X, α Y + Z$	7	$X - Z$
4	$α X + Z, Y$	8	$X$

表2 子李代数L3的最优化系统

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