微分方程与广义函数的群分析
Group analysis of differential equations and generalized functions
摘要：
我们提出了一个扩展的经典李群分析方法的微分方程方程涉及广义函数（特别是：分布）。Colombeau的广义函数代数论为这种推广提供了一个合适的框架。我们证明了在一些温和的条件下的微分方程，经典解的对称性保持对称的广义解。此外，我们介绍了一个推广的无穷小群分析方法，允许计算对称性的线性和非线性微分方程包含广义函数条款。由此，群生成元和群作用可以由广义函数本身给出。
英文摘要：
We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the infinitesimal methods of group analysis that allows to compute symmetries of linear and nonlinear differential equations containing generalized function terms. Thereby, the group generators and group actions may be given by generalized functions themselves.
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