Closing Comments

Symmetries of differential equations are useful in understanding their solutions and as a way to understand the techniques that we use to find the solutions. We have shown how some of these techniques can be derived from the symmetries of the equations and how generalizations of the methods for solving ordinary differential equations lead to exact solutions for a couple of partial differential equations.

To assist in determining the symmetries for equations, we have also introduced a package MathSym that performs many of the calculations. In addition to the calculation of the Lie symmetries for differential equations that we have demonstrated here, MathSym can use ideas from differential Gröbner bases to generate a reduced form of the determining equations. For a discussion of algorithms similar to the ones that we have used, see [9].

MathSym can also generate the determining equations for the conditional symmetries first introduced by Bluman and Cole in 1969 [13]. Reductions with respect to conditional symmetries include the reductions derived using the direct method of Clarkson and Kruskal [14].

Finally, I do not consider MathSym a finished product. There are generalizations of Lie's technique that the package does not currently address. A lot of recent work in symmetry techniques for differential equations focuses on generalized symmetries and symmetries of difference equations. It is my intention to incorporate some of these ideas into future versions of the MathSym package.