The MathLie Project

The objective of MathLie is to efficiently combine computer algebra with solution strategies for differential equations. The advantage of MathLie is to obtain analytic solutions of partial as well as ordinary differential equations in split seconds. MathLie revives an excellent theory by Sophus Lie and makes available a large number of strategies to solve differential equations.

The method of symmetry analysis established by Sophus Lie is an algorithmic but time-consuming procedure to find solutions for differential equations (DEs). Today, this difficulty is solved by using computer algebra systems (CAS) such as Macsyma, Maple, Mathematica, or Reduce. Unfortunately, these standard systems do not support symmetry methods. Consequently it is necessary to extend CAS to point symmetries, generalized symmetries, potential symmetries, and approximate symmetries.

The packages used in MathLie are a centered hierarchy of basic functions. The inner most levels contain data objects and basic functions used in outer levels.

The package LieBasic defines low level functions and data objects needed by other packages. The packages Matrix and Generator provide corresponding objects as well as appropriate functions for manipulating the basic functions consequently all higher level packages, in particular LieAlgebra, is able to handle all objects in a unique way. The structure is designed in such a way it can be extended any other objects, e.g. spinors etc. The package DetEqus calculates the determining equations with options for point symmetries as well as generalized symmetries, potential symmetries, non-classical symmetries and approximate symmetries. Finally, the packages LieSolveODE and  LieSolvePDE provide functions for integrating determining equations, determining similarity reductions and solutions of differential equations. The inner packages of MathLie make up a complete Lie tool.

For computer systems with graphical user interface like X or Windows, there is an additional Graphical User Interface (GUI) providing standard mathematical notations. The master package MathLie loads the package GUI and any other package as soon as a built-in function is called.

We are currently working on extensions for generalized symmetries, first- and higher-order ODEs, algebraic structures and optimal systems of sub-algebras.

Created by Mathematica  (September 15, 2003)