Melting and vaporization of solids occur very often in nature and in man-made processes. Many analytical and numerical solutions exist for solving the temperature field in the liquid and solid regions, but inaccuracies persist in tracking the phase change interfaces and the numerical solution of the temperature field is usually cumbersome. The Boundary Element Method is proposed as an accurate, efficient way to solve for the temperature field and the interface positions in a phase change problem involving combined melting and vaporization. When comparing to specific one-dimensional test cases, accurate results arc obtained when using a sufficiently small time step. A comparison is made to existing data from a laser drilling experiment. The anticipated physical effects which occur on semi-infinite and finite domains arc confirmed. Consequently, this method can be used to model natural and industrial phenomena involving phase change. / Master of Science