Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 31, Issue 6, Pages 1192-1213Publisher
SIAM PUBLICATIONS
DOI: 10.1137/S003614109833450X
Keywords
algebras of generalized functions; Lie symmetries of differential equations; group analysis; delta waves; Colombeau algebras
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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 us 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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