Differential Equation Solvers & Tools
PDELIE is a Macsyma package developed by Peter Vafeades from
Trinity University. It comes with Macsyma in the share library.
This package analyzes differential equation systems using
Lie symmetry group methods. The analysis starts by determining the
symmetry vectorfields for the particular differential system, be they
geometric or of Bessel-Haagen type. The user may then compute structure
constants for Lie subalgebras generated by the symmetry vectorfields.
Lie algebras or subalgebras may then be used to reduce the differential
system to one involving fewer dependent and independent variables. This
usually leads to either an ordinary differential equation or an
algebraic system. The reduced equations sometimes can be solved to give
explicit symbolic solutions for the original system of equations. When
the system is variational, the user may compute the Noether conservation
laws of the system.
- AUTOMATIC COMPUTATION OF SYMMETRY VECTORS: PDELIE proceeds unassisted
by the user to compute the complete set of symmetry vectors for the
system in many cases.
- GENERALIZED SYMMETRY COMPUTATION: PDELIE automatically solves for
Bessel-Haagen generalized symmetries of the order selected by the user
through the PL_EOR global option variable.
- ALGEBRAIC CONSTANTS: When the system involves simple algebraic
parameters, PDELIE computes symmetries and similarity solutions.
- LIE ALGEBRAS: The structure constants of Lie subalgebras generated
by the symmetry vectorfields can be computed using PDELIE.
- DIFFERENTIAL INVARIANTS: The differential invariants for symmetry
vectors are computed automatically.
- SIMILARITY ANALYSIS AND SIMILARITY SOLUTIONS: The differential
invariants may be used to reduce the differential solution. PDELIE
computes the similarity solutions to a great number of systems of
ordinary as well as partial differential equations of many types,
linear as well as nonlinear, coupled or uncoupled.
- EULER OPERATORS: PDELIE can compute the result of applying regular
or extended Euler operators to an expression.
- CONSERVATION LAWS: for variational systems, the Noether conservation
laws and the generalized or Bessel-Haagen conservation laws may also
- When trigonometric functions play a significant role in the solution
of the determining system, results are less than impressive.
- Problems must be in Cartesian coordinates.
- PL_SOLVE uses the invariant form method which depends on MACSYMA's
ode solvers to come up with differential invariants. The direct
substitution method may be used when the invariant form method fails
but this method has not yet been fully implemented in this package.
- PL_SOLVE requires that symmetry vectors have no more than one nonzero
coefficient. Similarity solutions for vectors with more than one
nonzero entry may be computed.
Work in Progress
Boundary Value Problems may be solved through Lie symmetry group methods.
In the case of linear systems, superposition makes this task easier; for
nonlinear systems, however, the differential system as well as the boundary
conditions and boundary of the domain must share the same symmetries. Since
this restriction is so severe, it would be useful if the package could
suggest the type of boundary conditions and the shape of domains in which
similarity solutions computed by PL_SOLVE would be valid.
Special Purpose Systems
Last updated: November 15, 1994