  Nonlinear boundary value problems (BVPs) by means of the classical Lie
symmetry method are studied. A new definition of Lie invariance for BVPs is
proposed by the generalization of existing those on much wider class of BVPs. A
class of two-dimensional nonlinear boundary value problems, modeling the
process of melting and evaporation of metals, is studied in details. Using the
definition proposed, all possible Lie symmetries and the relevant reductions
(with physical meaning) to BVPs for ordinary differential equations are
constructed. An example how to construct exact solution of the problem with
correctly-specified coefficients is presented and compared with the results of
numerical simulations published earlier.
