A new procedure for computing the delay Lyapunov matrix for periodic time-delay systems that is based on the numerical solution of a partial differential equations (PDE) system is presented. The introduction of a new set of boundary conditions that are satisfied by the PDE system allows us to propose a new methodology for computing the initial conditions required by the implemented numerical scheme. The potential of the presented results is demonstrated by obtaining robust stability conditions depending on the delay Lyapunov matrix with respect to the system parameters, the delay and the frequency. The theoretical results are applied to the widely known delayed Mathieu equation.