The Laplace transformation and expansions into eigenfunctions have been used for a long time in applied mathematics. See, for example, Carrier and Pearson (1976). Our aim was mainly to show that the energy method is rather restrictive with respect to the admissable boundary conditions, though it is very powerful when it applies.

The proof of Theorem 7.8.1 is conceptually rather simple. We obtain immediately the basic energy estimate. Then, in the same way as for parabolic equations, we can estimate the time derivatives. This provides bounds for the x-derivatives. Corresponding estimates for suitable difference approximations (Kreiss (1960)) can be obtained in a similar way, and the existence of a solution follows as in the text.