您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2025, Vol. 60 ›› Issue (8): 125-134.doi: 10.6040/j.issn.1671-9352.0.2023.412

• • 上一篇    

一类(3+1)维宏观群体平衡方程的矩方法和尺度变换群分析及自相似解

林府标,杨洋   

  1. 贵州财经大学数学与统计学院, 贵州 贵阳 550025
  • 发布日期:2025-07-25
  • 作者简介:林府标(1978— ),男,副教授,博士,研究方向为偏微分方程. E-mail:linfubiao0851@163.com
  • 基金资助:
    国家自然科学基金资助项目(12461038);贵州省科技计划基金资助项目(黔科合基础-ZK［2022］一般021);贵州省高等学校系统建模与数据挖掘重点实验室基金资助项目(2023013);贵州财经大学创新探索及学术新苗项目(2024XSXMB16)

Method of moments, scaling transformation group analysis and self-similar solutions of a class of 3+1-dimensional macroscopic population balance equation

LIN Fubiao, YANG Yang   

  1. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, Guizhou, China
  • Published:2025-07-25

PDF (PC)

5

摘要： 探究一类带齐次碰撞核的(3+1)维宏观群体平衡(积分偏微分)方程的解析解法及自相似解。采用矩方法将(3+1)维积分偏微分方程转化成(2+1)维偏微分矩方程。利用尺度变换群方法获得了(3+1)维积分偏微分方程和(2+1)维偏微分矩方程的尺度变换群、约化的(2+1)维积分偏微分方程、约化的(1+1)维常微分方程、自相似解、精确解,分析解的动力学性态。

关键词: 宏观群体平衡方程, 矩方法, 尺度变换群, 自相似解

Abstract: Self-similar solutions and approaches for analytically solving a class of(3+1)-dimensional macroscopic population balance(integro-partial differential)equations with homogeneous collision kernels are studied.(3+1)-dimensional integro-partial differential equation is successfully transformed into(2+1)-dimensional partial differential moment equations by use of the method of moments. The admitted scaling transformation groups, reduced(2+1)-dimensional integro-partial differential equations, reduced(1+1)-dimensional ordinary differential equations, self-similar solutions, explicit exact solutions of(3+1)-dimensional integro-partial differential equations and(2+1)-dimensional partial differential moment equations are presented by use of the method of scaling transformation analysis. The dynamic behaviour analysis of the obtained solutions are also given.

Key words: macroscopic population balance equation, moment method, scaling transformation group, self-similar solution

中图分类号: 

  • O175.6
[1] RAMKRISHNA D. Population balances, theory and applications to particulate systems in engineering[M]. San Diego: Academic Press, 2000.
[2] CAMERON I T, WANG F Y, IMMANUEL C D, et al. Process systems modelling and applications in granulation, a review[J]. Chemical Engineering Science, 2005, 60:3723-3750.
[3] PODGÓRSKA W. Multiphase particulate systems in turbulent flows-fluid-liquid and solid-liquid dispersions[M]. New York: CRC Press, 2020.
[4] RANDOLPH A D, LARSON M A. Theory of particulate processes: analysis and techniques of continuous crystallization[M]. San Diego: Academic Press, 1988.
[5] WANG Kaiyuan, YU Suyuan, PENG Wei. Extended log-normal method of moments for solving the population balance equation for Brownian coagulation[J]. Aerosol Science and Technology, 2019, 53(3):332-343.
[6] RAMKRISHNA D, SINGH M R. Population balance modeling: current status and future prospects[J]. Annual Review of Chemical and Biomolecular Engineering, 2014, 5:123-146.
[7] 林府标,张千宏. 一类细胞分裂群体平衡方程的对称群及精确解[J]. 四川师范大学学报(自然科学版),2021,44(2):209-215. LIN Fubiao, ZHANG Qianhong. Symmetry groups and exact solutions of a class of population balance equations with cells fission processes[J]. Journal of Sichuan Normal University(Natural Science), 2021, 44(2):209-215.
[8] 林府标,张千宏. 纯粹的破损过程的群体平衡方程的对称及精确解[J]. 杭州师范大学学报(自然科学版),2021,20(3):295-303. LIN Fubiao, ZHANG Qianhong. Symmetries and exact solutions of population balance equation with pure breakage processes[J]. Journal of Hangzhou Normal University(Natural Science Edition), 2021, 20(3):295-303.
[9] 林府标,杨欣霞,张千宏. 一类带齐次凝聚核的Smoluchowski方程的尺度变换群及相似解[J]. 西北师范大学学报(自然科学版),2022,58(3):18-24. LIN Fubiao, YANG Xinxia, ZHANG Qianhong. Scaling transformation group and similarity solutions for a class of Smoluchowski equation with homogeneous coagulation kernels[J]. Journal of Northwest Normal University(Natural Science), 2022, 58(3):18-24.
[10] 林府标,张千宏. 用伸缩变换群分析法探究群体平衡方程的自相似解[J]. 应用数学学报,2020,43(5):833-852. LIN Fubiao, ZHANG Qianhong. Self-similar solutions of the population balance equation, using scaling group analysis[J].Acta Mathematicae Applicatae Sinica, 2020, 43(5):833-852.
[11] 林府标,杨欣霞,张千宏. 一类积分-偏微分群体平衡方程的非完全不变群及显式精确解[J]. 华南师范大学学报(自然科学版),2023,55(5):72-79. LIN Fubiao, YANG Xinxia, ZHANG Qianhong. Incomplete invariant groups and explicit exact solutions of a class of integro-partial differential population balance equation[J]. Journal of South China Normal University(Natural Science Edition), 2023, 55(5):72-79.
[12] 林府标,张千宏. 一类群体平衡方程的伸缩变换群分析及自相似解[J]. 安徽大学学报(自然科学版),2020,44(3):6-12. LIN Fubiao, ZHANG Qianhong. Analysis of scaling transformation group and self-similar solutions of a class of population balance equation[J]. Journal of Anhui University(Natural Science Edition), 2020, 44(3):6-12.
[13] 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. LIN Fubiao, ZHANG Qianhong. Analysis of scaling transformation group and explicit exact solutions of the population balance equation involving breakage and growth processes[J]. Journal of Zhejiang University(Science Edition), 2022, 49(1):36-40.
[14] 蔡振宁,樊玉伟,胡志成,等. 矩方法在动理学中的发展与应用[J]. 中国科学(信息科学),2016,46(10):1465-1488. CAI Zhenning, FAN Yuwei, HU Zhicheng, et al. The development and application of the moment method in the gas kinetic theory[J]. Scientia Sinica(Informationis), 2016, 46(10):1465-1488.
[15] NONNENMACHER T F. Application of the similarity method to the nonlinear Boltzmann equation[J]. Journal of Applied Mathematics and Physics(ZAMP), 1984, 35:680-691.
[16] MELESHKO S V. Methods for constructing exact solutions of partial differential equations:mathematical and analytical techniques with applications to engineering[M]. New York: Springer, 2005.
[17] GRIGORIEV Y N, IBRAGIMOV N H, KOVALEV V F, et al. Symmetries of integro-differential equations: with applications in mechanics and plasma physics[M]. New York: Springer, 2010.
[18] LIN F B, MELESHKO S V, FLOOD A E. Exact solutions of the population balance equation including particle transport, using group analysis[J]. Communications in Nonlinear Science and Numerical Simulation, 2018, 59:255-271.
[19] IBRAGIMOV N H, KOVALEV V F, PUSTOVALOV V V. Symmetries of integro-differential equations: a survey of methods illustrated by the Benny equations[J]. Nonlinear Dynamics, 2002, 28:135-153.
[1] 李希亮1,2，韩玉良2. 一类半线性积分-微分方程几乎自守温和解的
存在惟一性和稳定性[J]. J4, 2013, 48(12): 30-34.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发