Abstract :
The nonclassical method of reduction was devised originally by Bluman and Cole
in 1969, to find new exact solutions of the heat equation. Much success has been
had by many authors using the method to find new exact solutions of nonlinear
equations of mathematical and physical significance. However, the defining equations
for the nonclassical reductions of the heat equation itself have remained
unsolved, although particular solutions have been given. Recently, Arrigo, Goard,
and Broadbridge showed that there are no nonclassical reduction solutions of
constant coefficient linear equations that are not already classical Lie symmetry
reduction solutions. Their arguments leave open the problem of what is the general
nonclassical group action, and its effect on the relevant solution of the heat
equation. In this article, both these problems are solved. In the final section we use
the methods developed to solve the remaining outstanding case of nonclassical
reductions of Burgers’ equation.
