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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| Format: | Article |
| Language: | English |
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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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| author | Ricardo Alfaro Jim Schaeferle |
| author_facet | Ricardo Alfaro Jim Schaeferle |
| author_sort | Ricardo Alfaro |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-88474704d2d049e899a49aebeb2a0bb5 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2004-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-88474704d2d049e899a49aebeb2a0bb52025-02-03T01:23:48ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252004-01-012004512741275310.1155/S016117120430904XCoefficients of prolongations for symmetries of ODEsRicardo Alfaro0Jim Schaeferle1Department of Mathematics, University of Michigan-Flint, Flint 48502, MI, USADepartment of Mathematics, University of Michigan-Flint, Flint 48502, MI, USASophus 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.http://dx.doi.org/10.1155/S016117120430904X |
| spellingShingle | Ricardo Alfaro Jim Schaeferle Coefficients of prolongations for symmetries of ODEs International Journal of Mathematics and Mathematical Sciences |
| title | Coefficients of prolongations for symmetries of ODEs |
| title_full | Coefficients of prolongations for symmetries of ODEs |
| title_fullStr | Coefficients of prolongations for symmetries of ODEs |
| title_full_unstemmed | Coefficients of prolongations for symmetries of ODEs |
| title_short | Coefficients of prolongations for symmetries of ODEs |
| title_sort | coefficients of prolongations for symmetries of odes |
| url | http://dx.doi.org/10.1155/S016117120430904X |
| work_keys_str_mv | AT ricardoalfaro coefficientsofprolongationsforsymmetriesofodes AT jimschaeferle coefficientsofprolongationsforsymmetriesofodes |