含源项的Smoluchowski方程的预李群分类

利用预李群分类法研究了带源函数和齐次核函数的非齐次积分—偏微分Smoluchowski方程的部分群分析.首先应用改进的李群分析法得到了带齐次核函数的齐次积分—偏微分Smoluchowski方程的对称、完全群分类和最优化子李代数系统. 其次进一步用预李群分类法获得了相应带齐次核函数的非齐次积分—偏微分Smoluchowski方程的决定方程、决定方程的通解、群不变解、显式解析解和约化的积分-常微分方程.最后所获得研究结果表明预李群分类法不但能用于偏微分方程而且也可应用于积分—偏微分方程.

Preliminary group classification of Smoluchowski equation with a source term

The partial group analysis of a nonhomogeneous integro-differential Smoluchowski equation with a source term and homogeneous kernel function is investigated by use of the method of preliminary group classification in this paper. Firstly， the symmetries， complete group classification and optimal systems of subalgebras of homogeneous integro-differential Smoluchowski equation with homogeneous kernel are obtained using the method of developed Lie group analysis. Secondly， the determining equation， general solutions to determining equation， invariant solutions and explicit analytical solutions for the corresponding nonhomogeneous integro-differential Smoluchowski equation with a source term and homogeneous kernel function are presented by the method of preliminary group classification. Finally， the obtained results have demonstrated that the method of preliminary group classification can be applied not only to partial differential equations， but also to integro-differential equations.