组分类的映射法
Mapping method of group classification
摘要：
我们重新审视整个框架的组分类的微分方程。在引入了微分方程弱相似类的概念后，我们发展了这类微分方程的群分类的映射方法，它推广了已有文献中的所有版本。本文将这种映射方法应用于各类Kolmogorov方程和空间维为1的Fokker-Planck方程的群分类。计算了这些类的等价群胚和等价群。这些类相对于相应的等价群的组分类问题减少到寻找所有不等价的热方程的不等价的解决方案与不等价的潜力承认李对称扩展。这种减少使我们能够彻底解决的类的Kolmogorov和Fokker-Planck方程与时间无关的系数的组分类问题。
英文摘要：
We revisit the entire framework of group classification of differential equations. After introducing the notion of weakly similar classes of differential equations, we develop the mapping method of group classification for such classes, which generalizes all the versions of this method that have been presented in the literature. The mapping method is applied to group classification of various classes of Kolmogorov equations and of Fokker-Planck equations in the case of space dimension one. The equivalence groupoids and the equivalence groups of these classes are computed. The group classification problems for these classes with respect to the corresponding equivalence groups are reduced to finding all inequivalent solutions of heat equations with inequivalent potentials admitting Lie-symmetry extensions. This reduction allows us to exhaustively solve the group classification problems for the classes of Kolmogorov and Fokker-Planck equations with time-independent coefficients.
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