International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 51, Pages 2741-2753
doi:10.1155/S016117120430904X
Coefficients of prolongations for symmetries of ODEs
Department of Mathematics, University of Michigan-Flint, Flint 48502, MI, USA
Received 4 September 2003
Copyright © 2004 Ricardo Alfaro and Jim Schaeferle. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Sophus Lie developed a systematic way to solve ODEs. He found that
transformations which form a continuous group and leave a
differential equation invariant can be used to simplify the
equation. Lie's method uses the infinitesimal generator of these
point transformations. These are symmetries of the equation
mapping solutions into solutions. Lie's methods did not find
widespread use in part because the calculations for the
infinitesimals were quite lengthy, needing to calculate the
prolongations of the infinitesimal generator. Nowadays,
prolongations are obtained using Maple or Mathematica, and Lie's
theory has come back to the attention of researchers. In general,
the computation of the coefficients of the (n)-prolongation is
done using recursion formulas. Others have given methods that do
not require recursion but use Fréchet derivatives. In this
paper, we present a combinatorial approach to explicitly write the
coefficients of the prolongations. Besides being
novel, this approach was found to be useful by the authors for
didactical and combinatorial purposes, as we show in the examples.