Demonstration of the REDUCE Package CRACK
for Investigating Partial Differential Equations
Thomas Wolf Thomas Wolf [twolf(at)brocku.ca] and Andreas Brand (Jena)
The CRACK package attempts to solve an
overdetermined system of ordinary or partial differential equations (ODEs/PDEs)
with at most polynomial nonlinearities.
The latest version is available from
A manual for CRACK is available in
pdf format as well as
a more detailed description with emphasis on some of the computer algebra algorithms
(LATEX documents will be properly formatted by
your web browser only if you have installed the
IBM techexplorer or
equivalent plug-in). This demonstration is based on the CRACK manual.
Applications of CRACK
Overdetermined systems of DEs most frequently occur when other
"differentiable objects", for example, other DEs or a metric of some
Riemannian space, are investigated concerning their properties in some open
neighbourhood (not discrete properties or properties at a single point).
Examples are the investigation of DEs concerning:
- Lie symmetries (point, contact or generalized symmetries);
- conservation laws;
- the use of ansδtze for finding
- an integrating factor for a DE,
- a Lagrangian equivalent to an ODE,
- a factorization of an ODE into a first order ODE plus lower order PDEs.
In these cases, typically another program is used to cast the mathematical
problem into an overdetermined system of PDEs and then call CRACK. Examples are the programs LIEPDE, CONLAW, LAGRAN, DECOMP to
investigate DEs and the program CLASSYM to find
Killing-vectors and -tensors for a given space-time metric.
As CRACK has a number of integration
facilities built in, it can also be used to solve non-overdetermined DE-systems
directly if they are not too difficult. An example is the program APPLYSYM, which takes as input Lie symmetries
(computed, for example using LIEPDE beforehand)
and uses QUASILINPDE to solve a linear
first-order PDE to find the symmetry and similarity variables. To solve
this PDE, CRACK is called with the corresponding
(non-linear, non-overdetermined, autonomous) characteristic ODE system.
A feature of CRACK that is not supported by
this web-based demonstration (mainly for reasons of security) is its ability to
be used interactively.
Original design by Francis Wright
and Thomas Wolf
Maintained by Thomas Wolf [twolf(at)brocku.ca]
Last updated 10 January 2006