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IMA Journal of Applied Mathematics Advance Access originally published online on July 11, 2005
IMA Journal of Applied Mathematics 2006 71(2):210-231; doi:10.1093/imamat/hxh094
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© The Author 2005. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A contribution to the symmetry classification problem for second-order PDEs in z(x, y)

Erik Hillgarter**

Research Institute for Symbolic Computation (RISC Linz), Johannes Kepler University, A-4040 Linz, Austria

** Email: hillgarter{at}risc.uni-linz.ac.at

I report on a contribution to the point symmetry classification problem for second-order partial differential equations (PDEs) in z(x, y), i.e. to an overview over all possible symmetry groups admitted by this class of equations. The article also contains a concise introduction into classical symmetry analysis.

Sophus Lie (1842–1899) determined all continuous transformation groups of the 2D plane and gave normal forms for any ordinary differential equation that is invariant under one of those groups. I deal with the extension of Lie's program to second-order PDEs in z(x, y). The starting point to this endeavour is a previously unknown paper by Amaldi from 1901, which claims to have completed Lie's classification of groups acting in (x, y, z)-space. I also present a Maple procedure (‘LHSO1_PDE_Solver’) for solving systems of linear, homogeneous first-order PDEs that performs better on this class than Maple's built-in PDE system solver.

Keywords: differential invariants; Lie groups; (partial) differential equations; symmetry analysis; symmetry group classification.

Received on 18 October 2004. accepted on 21 February 2005.


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