Projective analysis and preliminary group classification of the nonlinear fin equation $u_t=(E(u)u_x{)}_x+h(x)u$

In this paper we investigate for further symmetry properties of the nonlinear fin equations of the general form $u_t=(E(u)u_x{)}_x+h(x)u$ rather than recent works on these equations. At first, we study the projective (fiber-preserving) symmetry to show that equations of the above class can not be reduced to linear equations. Then we determine an equivalence classification which admits an extension by one dimension of the principal Lie algebra of the equation. The invariant solutions of equivalence transformations and classification of nonlinear fin equations among with additional operators are also given.