Science, Technology and Engineering

Symmetry Determination and Linear Differential Equation Package

Examples

A guide to the examples

The data files in the examples directory consist of some easy problems and some very hard problems, but almost all have been included to illustrate strategies for how to handle problems which don't go through Dimsym as easily as heat1, either because there is more to specifying the problem or because we need to coax Dimsym through to the solution. Many of the data files have references to where the problem came from. The examout directory contains the output generated by the files in the examples directory. The directory headfiles contains the datafiles circulated by Alan Head with his program LIE, modified for use with Dimsym. They are a good set of test files, and show Dimsym happily performing its duties. The headout directory contains the output generated by the files in the headfiles directory.

Brief desciptions of the example files follow.

• blas, blasf, blasf1, blasf2
  These are variations of the same problem, looking for symmetries of the third order blasius equation expressed as a system of first order equations. These examples serve to show the importance of placing the original differential equations in Standard Form.
• body3_3d
  This file is an example of the power of Dimsym and some of the tricks I use to coax Dimsym through a problem of this magnitude. The problem being considered is the general three body problem in 3-space, that is, the newtonian dynamics of three point masses (or charges) under the mutual attraction of conservative radial forces. So it is a system of 9 second order O.D.E.s.
• burglb3, burglb3a, burglb3b
  These files are all variations of the same problem, looking for third order Lie-Backlünd symmetries of Burgers equation.
• cent3d & cent3da
  Here we take two slightly different approaches to the central force problem in cartesian coordinates. This is an elegant example of a classification problem and the use of subexpressions, not entirely unlike body3_3d except that it is a much simpler problem to solve.
• cev
  This file shows two more features of Dimsym. We set up the equation and look for point symmetries as usual; however there are two determining equations remaining after solvedets(std) is finished.
• exsho, exshofi
  These files demonstrate interaction between Dimsym and EXCALC, and how to use Dimsym to solve equations other than those formed by mkdets(). This example examines the simple harmonic oscillator.
• feder
  This is an example of nonlocal symmetry of the Federbush model, reproducing the results of Kersten.
• fpu, fpu1 & fpu2
  These are different ways of looking for higher order symmetries of the Fermi-Pasta-Ulam equation. In fpu, neither std nor stdform1 are able to solve the determining equations, and both grind on for a long time and lead to expression swell. In fpu1 we express the second order P.D.E. as a system of first order equations and obtain a solution. In fpu2 certain redundancies are removed from the outset and obtain a solution.
• heat1
  This is just the standard example for symmetry analysis.
• hilbcart<, hilbcart1 & hilbcart2
  These all calculate the 14 internal symmetries of the Hilbert-Cartan equation. The difference between hibcart and hibcart1 is the choice of leading derivative. An interesting way of looking at this problem is to use a freeunknown for the sign of the square-root, as done in hilbcart2, which makes it easier to compare the two algebras in the two different halfspaces.
• igf1
  This example of finding the point symmetries of isentropic gas flow solves the determining equations much faster if we place the equations in standard form first.
• jm
  A test file.
• karpman
  Dimsym makes short work of these equations. Compare the time (88s cpu on an IBM 6000) with the 3 hours cpu on a VAX 8650 for the MACSYMA program SYMMGRP.MAX reported in [CHW91], which leaves 69 determining equations to be solved manually (via the usual feedback process).
• liouv3
  This example shows an interesting feature of Dimsym which lets us keep explicit integrals of explicit or freeunknown expressions.
• mhd, mhd1
  This example comes from [Her93], with a reported time of 50 minuites cputime just to create the 222 determining equations which must then be solved manually, using the feedback process. Reid and Wittkopf [RW93] report around 1 hour cpu to place these equation in Standard Form. Dimsym solves the whole problem automatically: forming the determining equations, solving them, and forming the generators, in less than 11 minuites cpu on an IBM 6000. The file mhd1 is just the same equations as mhd, with a different resolution.
• reid1
  Data file reid1 contains the example from [RB91]. We form the determining equations as usual and use solvedets stdsplit to get the equations which are usually referred to as the determining equations. This gives us nine equations which do not look easy to solve, but once we use solvedets stdform> to place the equations in standard form (without any integration), we are left with nine easy equations which only require a few simple integrations to solve.
• subic, subic1
  Data file subic shows how we can ``lose'' symmetries if we do not properly include all necessary integrability conditions. In subic1 this is rectified and the extra symmetries are found.
• sho
  This example demonstrates the use of the odeslv algorithm. It utilises the simple harmonic oscillator.
• ure, ure1
  Here another important aspect of giving the equations in Semi-Standard Form is illustrated. The data file ure finds the point symmetries of Ure's equations, no problem. The data file ure1 contains Ure's equations, but this time they are not given in an allowable resolution, leading to an infinite recursive substitution.

References

1. [CHW91] B. Champagne, W. Hereman and P. Winternitz, The Computer Calculation of Lie Point Symmetries of Large Differential Equations., Computer Physics Communications 66, 319-340.
2. [Hea93] A. K. Head, LIE: A PC Program for Lie Analysis of Differential Equations, Computer Physics Communications 77, 241-248.
3. W. Hereman, Review of Symbolic Software for Computation of Lie Symmetries of Differential Equations, Euromath Bulletin 2.
4. [RB91] G. J. Reid and A. Boulton, Reduction of Systems of Differential Equations to Standard Form and their Integration using Directed Graphs Without Integrating Determining Equations, Proceedings of the International Symposium on Symbolic and Algebraic Computation, Bonn.
5. [RW93] G. J. Reid and A. D. Wittkopf, Long Guide to the Standard Form Package, Preprint, Mathematics Department, University of British Columbia, Vancouver, Canada.