Foundations of Computational Mathematics

, Volume 6, Issue 3, pp 353–386 | Cite as

Symmetry Classification Using Noncommutative Invariant Differential Operators

Article

Abstract

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group Gf or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined "defining system" of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations vx = u, vt = B(u) ux - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.

Lie symmetries of PDEs Equivalence group Symmetry classification of PDE Moving frames Noncommutative invariant differential operators Reduced involutive form Diffusion convection equation 

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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.School of Information Sciences and Engineering, University of Canberra, Canberra, ACT 2600Australia
  2. 2.Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7Canada

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