One-Dimensional Fourth-Order Boundary-Value Problems

In all the problems we have considered thus far, the highest derivatives appearing in the governing differential equations are of order two. The corresponding weak forms then contain derivatives of first order, and therefore it is only necessary that the trial and test functions be in \(H ^1\) in order to be admissible. Other problems in mathematical physics, however, involve higher-order derivatives. For instance, the deformation of plates and shells entails fourth and even higher-order derivatives.