Group analysis of differential equations and generalized functions

Details Group analysis of differential equations and generalized functions Group analysis of differential equations and generalized functions

1999-12-28

arxiv.org/abs/math/9912217v1

SIAM J. Math. Anal. 31 (2000) no. 6, 1192-1213.

Mathematics

Functional Analysis

27 pages, LaTeX

Scientific paper

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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