Lie group symmetries as integral transforms of fundamental solutions

We obtain fundamental solutions for PDEs of the form u(t) = sigma x(gamma)u(xx) + f(x)u(x) - mu x(r)u by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be reduced to inverting a Laplace transform or some other classical transform. (c) 2006 Elsevier Inc. All rights reserved.