A fixed-grid local method for 1-D Stefan problems

This note is concerned with a fixed grid local method for the solution of 1-D free boundary problems. The method uses a coordinate transformation. After the coordinate transformation, it is then possible to obtain exact solutions for the resulting equations in terms of the local variables. These exact solutions are in terms of constants of integration that are unknown. By imposing the given boundary conditions, and some smoothness requirements, it is possible to furnish a set of linearly independent conditions that can be used to solve for the constants of integration. A number of examples are used to study the applicability of the method.