Demonstration of the REDUCE Package ConLaw for Investigating Conservation Laws of Partial Differential Equations and First Integrals of Ordinary Differential Equations
ConLaw: Thomas Wolf
The ConLaw package attempts to compute first integrals for single ordinary differential equations (ODEs) or systems of ODEs, and conservation laws for single partial differential equations (PDEs) or systems of PDEs.
A 5-page manual for ConLaw is available in tex, dvi, ps, pdf format as well as a more detailed description with emphasis on some of the computer algebra algorithms [dvi, ps, pdf] (J. Symb. Comp. 27 (1999), 221-238) and a general description of the pro's and con's of the four different methods implemented in ConLaw1,...,ConLaw4 for finding conservation laws [dvi, ps, pdf] (Euro. J. of Applied Mathematics, 13, part 2 (2002) 129-152). (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 ConLaw manual.
The ConLaw package tries to find conservation laws for a given differential equation (ODE or PDE) or system of ODEs or PDEs of the form
ConLaw comes in four versions:
ConLaw1 tries to find the conserved current Pi by directly solving
ConLaw3 tries to find Pi and characteristic functions Qn by solving
Applying the Euler operator (variational derivative) for each ub on (3) gives a zero left hand side and therefore conditions involving only Qn. ConLaw4 tries to solve these conditions directly in all derivatives of ub and to compute Pi afterwards.
ConLaw2 does substitutions based on (1) before solving these conditions for Qn and therefore computes adjoined symmetries. If possible, the adjoined symmetries are completed to conservation laws by computing Pi from the Qn. (For more details please see the documentation).
All four procedures have the same syntax. They have two parameters, both of which are lists. The first parameter specifies the equations (1), the second specifies the computation to be done. One can either specify an ansatz for Pi and Qn or investigate a general situation by specifying only the order of the characteristic functions or the conserved current.
Original design by Francis Wright and Thomas Wolf Maintained by Thomas Wolf [twolf(at)brocku.ca] and Winfried Neun Last updated 23 January 2006