University of Kent at Canterbury

CATHODE Demonstration of
diffgrob2,
a Maple Package for Simplifying
Systems of Partial Differential Equations

 

DEMONSTRATION  UNAVAILABLE

 

This package was written by E. L. Mansfield.
The source code and further information about diffgrob2 are available via FTP, in particular:

Go to the home page for the UK CATHODE 2 Project.

Synopsis

The purpose of the package is to render tractable the analysis and perhaps integration of an overdetermined system of partial differential equations (PDEs). Implemented is a differential analogue of Buchberger's algorithm for computing a Gröbner basis of a system of polynomials.

Output location

This section is experimental and may not work correctly!

By default, the output will go to a new browser window. The output window may not be automatically brought to the front except when it is first created. It may be best to tile the input and output windows.

Alternatively, you can split one window into two frames:

input output
(Recommended for a large screen and maximized browser window.)
input
output
(Recommended for a smaller screen or non-maximized browser window.)
No frames (The default. Recommended for a very small screen and unavoidable if your browser does not support frames!)

However, frame output currently seems to work properly only with Microsoft Internet Explorer.

Select an operation

diffgrob2 provides the following main functions, most of which can be selected for demonstration. The default is the Kolchin-Ritt algorithm. (Clicking the R button will run the selected function on the example input shown, unless a problem has been explicitly input below.)

Name Description Example
Dep Ind PDEs
diffSpoly calculates the cross-derivative of two PDEs u x, y diff(U,x$2)^2*diff(U,y) + diff(U,x$2)*U - diff(U,x,y), diff(U,y$2)
reduce yields the pseudo-reduction of a PDE (the first) with respect to a given system of PDEs (the rest) u x, y -2*diff(U,x$2)*diff(U,y)*diff(U,x$2,y) - diff(U,x$2)^2*diff(U,y$2) - diff(U,x$2,y)*U - diff(U,x$2)*diff(U,y),
diff(U,x$2)^2*diff(U,y) + diff(U,x$2)*U - diff(U,x,y), diff(U,y$2)
reduceall pseudo-reduces each member of a set of PDEs with respect to the other members, until no further reductions are possible h x, t diff(H,x) + H^2, diff(H,x$2)
orthreduceall similar to reduceall, but with strict differential reduction used instead of pseudo-reduction h x, t diff(H,x) + H^2, diff(H,x$2)
KolRitt performs the Kolchin-Ritt algorithm on the given system of PDEs with respect to the given term ordering. Yields a differential Gröbner basis, or Standard Form, for linear and orthonomic systems. u x, y, z diff(U,z$2)-y*diff(U,x$2), diff(U,y$2)
  diffGbasis yields a differential Gröbner basis for a larger range of inputs than KolRitt u, f(U) r, s diff(U,r)*diff(U,s)-1, diff(U,r,s)-F

Optionally change the term ordering

Term orderings are explained on page 17 of the user guide.

The orderings lex, rlex, etc. depend on the ordering of the independent variables, which is the order in which they are listed in the input, e.g. input order x, y, z implies term ordering x < y < z.

Optionally enter a problem

If no problem is entered here then the example in the table of operations above will be run instead.

The input differential expressions are always displayed before the output, in the form

Input = ...
Output = ...
Factorized Output = ...

Enter the variables

Dependent: DepVars (e.g. u, v)
Independent: IndVars (e.g. x,y,z)

Automatically set up assignments of the form U := u(x,y,z), regardless of the case used for the dependent variables. You can then use whichever side of these assignments you prefer in setting up the rest of the problem. (This facility is illustrated by the examples in the table of operations above.)

Enter the equations

Enter a comma-separated sequence of Partial Differential Equations (PDEs). They must be differential expressions (not equations) in standard Maple syntax, e.g.
diff(U,z$2)-y*diff(U,x$2), diff(U,y$2)

Specify the output format and submit

The diffgrob2 output format is uninterpreted TEX using subscripts to denote differentiation. By default, this demonstration uses normal diffgrob2 output format and only the input and output are displayed. Alternatively ...

Interpret the TEX output (primarily for use with the IBM techexplorer or equivalent plug-in).
Not available if transcript output is selected below.

Use "d/dx" notation to denote differentiation and Maple two-dimensional character-based prettyprinted output.
Not available if interpreted TEX output is selected above.

Return a full transcript of the session showing Maple input as well as output.

The Maple output will be sent to another browser window or frame after you press this button …

… unless you change your mind!

This demonstration currently uses Maple V Release 4 running under Windows NT 4.0. The version of diffgrob2 being demonstrated was built from the UNIX source code file src_UNIX_cat, with some very minor modifications to support the I/O requirements of the demonstration.

Liz Mansfield and Francis Wright, 13 August 2004