We present a method for the solution of integral inequalities in one dimension that relies on the the solution of a system of coupled differential equations. We illustrate the proposed method in the problem of stability analysis of linear Partial Differential Equations (PDEs) by solving Lyapunov inequalities that guarantee exponential stability of the systems under consideration. 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 efficacy of the proposed numerical methodology using examples of different types of PDEs.