# PDELIE

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.

## Highlights

• 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 be computed.

## Limitations

• 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

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