Coefficients of prolongations for symmetries of ODEs
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 o...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2004-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S016117120430904X |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |