Canonical transforms, separation of variables, and similarity solutions for a class of parabolic differential equations

Using the method of canonical transforms, we explicitly find the similarity or kinematical symmetry group, all ’’separating’’ coordinates and invariant boundaries for a class of differential equations of the form [α∂2/∂q2+βq ∂/∂q+γq2+δq+ε∂/∂q+ζ] u (q,t) =−i (∂/∂t) u (q,t),or of the form [α′ (∂2/∂q2+μ/q2)+β′q∂/∂q+γ′q2] u (q,t) =−i (∂/∂t) u (q,t), for complex α,β,..., γ′. The first case allows a six-parameter WSL(2,R) invariance group and the second allows a four-parameter O(2) ⊗ SL(2,R) group. Any such differential equation has an invariant scalar product form which, in the case of the heat equation, appears to be new. The proposed method allows us to work with the group, rather than the algebra, and reduces all computation to the use of 2×2 matrices.