A solution to the problem of invariants for parabolic equations

The article is devoted to the solution of the invariants problem for the one-dimensional parabolic equations written in the two-coefficient canonical form used recently by N.H. Ibragimov:ut-uxx+a(t,x)ux+c(t,x)u=0.ut-uxx+a(t,x)ux+c(t,x)u=0.A simple invariant condition is obtained for determining all equations that are reducible to the heat equation by the general group of equivalence transformations.The solution to the problem of invariants is given also in the one-coefficient canonicalut-uxx+c(t,x)u=0.ut-uxx+c(t,x)u=0.One of the main differences between these two canonical forms is that the equivalence group for the two-coefficient form contains the arbitrary linear transformation of the dependent variable whereas this group for the one-coefficient form contains only a special type of the linear transformations of the dependent variable.