一类群体平衡方程的李群分析及精确解

林府标; 张千宏

doi:10.3969/j.issn.1000-5641.201911008

一类群体平衡方程的李群分析及精确解

doi: 10.3969/j.issn.1000-5641.201911008

贵州财经大学数统学院, 贵阳　550025

基金项目: 国家自然科学基金(11761018, 11361012); 贵州省科技计划基金项目(黔科合基础[2019]1051); 贵州省教育厅青年科技人才成长项目(黔教合KY字[2017]150); 2018年度贵州财经大学校级科研基金项目(2018XYB04)

详细信息

作者简介:
林府标, 男, 博士, 讲师, 研究方向为李群分析法在微分方程中的应用. E-mail: linfubiao0851@163.com

中图分类号: O175.5; O175.6
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出版历程
- 收稿日期: 2019-02-03
- 刊出日期: 2020-03-01

Lie group analysis and exact solutions for a class of population balance equations

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

摘要

摘要: 探求一类群体平衡方程的显式精确解. 首先将群体平衡方程转化成偏微分方程, 利用经典李群分析法获得了偏微分方程的对称, 进而得到了群体平衡方程的对称、最优化子李代数系统、约化的常微分-积分方程、群不变解及精确解. 其次采用试探函数法找到了约化的常微分-积分方程的部分精确解, 最后得到了群体平衡方程的部分显式精确解.
- 群体平衡方程 /
- 李群分析法 /
- 试探函数法 /
- 群不变解 /
- 精确解
Abstract: In this paper, exact solutions for a class of population balance equations were investigated. The population balance equation was first transformed to a partial differential equation; symmetries of the partial differential equation were then obtained by use of the classical Lie group analysis method. In addition, the paper presents symmetries, optimal system of subalgebras, reduced ordinary differential integral equations, and group invariant solutions of the population balance equation. Exact solutions of the reduced ordinary differential integral equations were subsequently found using trial functions. Finally, exact solutions for a class of population balance equations are derived.
- population balance equation /
- Lie group analysis method /
- method of trial function /
- group invariant solution /
- exact solution

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表 1 算子(11)的换位子表

Tab. 1 The table of commutators for the operators (11)

	$X_{1}$	$X_{2}$	$X_{3}$	$X_{4}$
$X_{1}$	$0$	$X_{1}+2X_{3}$	$- X_4$	$0$
$X_{2}$	$-X_{1}-2X_{3}$	$0$	$X_{3}$	$0$
$X_{3}$	$X_4$	$-X_{3}$	$0$	$0$
$X_{4}$	$0$	$0$	$0$	$0$

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表 2 实数域上李代数$ L_{4} $的最优化子李代数系统

Tab. 2 The optimal system of subalgebras for Lie algebra $ L_{4} $ over $ {\mathbb{R}} $

序号	子李代数	序号	子李代数
1	$ X_{1},\: X_{2},\: X_{3},\: X_{4} $	9	$ X_{2},\: X_{1}+X_{3} $
2	$ X_{1}+X_3,\: X_{2},\: X_{4} $	10	$ X_{2} $
3	$ X_{2},\: X_{3},\: X_{4} $	11	$ X_{1} $
4	$ X_{1},\: X_{3},\: X_{4} $	12	$ X_{4}+\alpha X_{3},\alpha\in{\mathbb{R}} $
5	$ X_{3},\: X_{4} $	13	$ X_{1}+X_{3} $
6	$ X_{4},X_{1}+\alpha X_{3},\alpha\in {\mathbb{R}} $	14	$ X_{1}-X_{3} $
7	$ X_{2}, \: X_{4} $	15	$ X_{3} $
8	$ X_{2},\:X_{3} $

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表 3 方程(5)的群不变解

Tab. 3 Invariant solutions of equation (5)

序号	子李代数	群不变解$ f(x,t) $	约化方程
1	$ X_{2} $	$ f(x,t) = \exp\big(-\frac{1}{2}t^2\big)\varphi(z),\:z = xt-t^2 $	(13)
2	$ X_{1} $	$ f(x,t) = \varphi(x) $	(14)
3	$ X_{4}+\alpha X_{3},\alpha\in {\mathbb{R}} $	$ f(x,t) = \exp\big(\frac{1-\alpha t}{\alpha}x\big)\varphi(t),\ t > \frac{1}{\alpha } $	(15)
4	$ X_{1}+X_{3} $	$ f(x,t) = \exp\big(-\frac{1}{2} t^{2}\big)\varphi(z),\ z = x-t $	(16)
5	$ X_{1}-X_{3} $	$ f(x,t) = \exp\big(\frac{1}{2} t^{2}\big)\varphi(z),\ z = x+t $	(17)
6	$ X_{3} $	$ f(x,t) = \exp(- t x)\varphi(t) $	(18)

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