Many important engineering problems Jail into the category of linear operators, with supporting boundary conditions. In this paper a new inner product and norm are developed that enable the numerical modeler to approximate such engineering problems by developing a generalized Fourier series. The resulting approximation is the “best” approximation in that a least-squares (L2) error is minimized simultaneously for fitting both the problem's boundary conditions and satisfying the linear operator relationship (the governing equations) over the problem's domain (both space and time). For slow-moving interface phase change problems where the heat flux balance can be adequately described by the Laplace equation, the generalized Fourier series technique results in a highly accurate solution.