The n-dimensional Laplace equation is of the form:

= + + ... + = 0

The determining systems for this equation are the simplest that we have posted Lie symmetry determining systems for.

The determining system consists of the dependent variables
and
*X*01, *X*02,..., *Xn* which depend on the independent variables
and
*x*01, *x*02,..., *xn*.

The systems for *n* = 3..16 can be downloaded
here
(the files in the archive are named laplace_n).
It should be emphasized that these are test systems only, as they are
homogeneous constant coefficient PDE in the infinitesimals,
and are relatively easy to bring to simplified form (as can be seen by the
run-times below).

Input | PureElim | |||

Dimension | Equations | Length | Time (sec) | Mem (MB) |

3 | 34 | 2723 | 0.01 | 0.70 |

4 | 58 | 5027 | 0.02 | 0.72 |

5 | 88 | 8211 | 0.04 | 0.75 |

6 | 124 | 12395 | 0.08 | 0.78 |

7 | 166 | 17699 | 0.15 | 0.86 |

8 | 214 | 24243 | 0.23 | 0.89 |

9 | 268 | 32147 | 0.39 | 1.00 |

10 | 328 | 41531 | 0.70 | 1.24 |

11 | 394 | 52515 | 1.10 | 1.39 |

12 | 466 | 65219 | 1.73 | 1.59 |

13 | 544 | 79763 | 3.19 | 1.88 |

14 | 628 | 96267 | 5.09 | 2.72 |

15 | 718 | 114851 | 8.77 | 3.43 |

16 | 814 | 135635 | 14.22 | 3.92 |

Allan Wittkopf 2000-06-15