A source-and-sink method has been developed for the solution of both ablation and combination of ablation and Stefan problems for a one-dimensional, subcooled medium imposed with both constant and time variant heat flux conditions. In this method the problem is solved in a fixed domain, so that the original heat flux condition imposed on the moving boundary is taken to be the condition imposed on the solid-vapor or liquid-vapor interface. This flux condition can then be used together with the temperature at the interface to solve for the interface position as well as the condition imposed on the fixed boundary. Finally, these interface position and boundary conditions are used in the temperature equation to complete the solution. The method has been used successfully in solving seven examples encompassing one-, two-, and three-phase ablation with the boundary of the medium imposed with constant, linear, and quadratic flux conditions. Numerical solutions for these examples as well as the trends of the moving boundary positions are discussed in detail.