We present a method for the stability analysis of a large class of linear Partial Differential Equations (PDEs) in one spatial dimension. We rely on Lyapunov analysis to establish the exponential stability of the systems under consideration. The proposed test for the verification of the underlying Lyapunov inequalities relies on the existence of solutions of a system of coupled differential equations. We illustrate the application of this method using a PDE actuated by a backstepping computed feedback law. Furthermore, for the case of PDEs defined by polynomial data, we formulate a numerical methodology in the form of a convex optimization problem which can be solved algorithmically. We show the effectiveness of the proposed numerical methodology using examples of different types of PDEs.