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Extension of Lie’s algorithm; a potential symmetries classification of PDEs

University of British Columbia

Symmetries of a system of differential equations are transformations which leave invariant the family of solutions of the system. Infinitesimal Lie symmetries of locally solvable analytic differential equations can be found by using Lie's algorithm. We extend Lie's algorithm to one which can calculate infinitesimal Lie symmetries of analytic systems of differential equations which are not locally solvable. Local infinitesimal symmetries of differential equations are flows of vector fields which depend on local properties of solutions and have been extensively calculated and applied. In contrast infinitesimal nonlocal symmetries, which are flows of vector fields depending on nonlocal properties of solutions, have only recently been introduced. Using our extension of Lie's symmetry algorithm, we study the infinitesimal nonlocal symmetries of potential type introduced by Bluman, Kumei and Reid. We give verifiable criteria for useful potential systems and give a complete potential symmetries analysis for a class of nonlinear diffusion equations. We also find large classes of higher order scalar and systems of partial differential equations admitting potential symmetries.

Thesis/Dissertation

application/pdf

2009-03-20

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http://hdl.handle.net/2429/6279

Mathematics

University of British Columbia

UBCV

DSpace