Separation of Variables for PDEs. A new look at an old subject.

Taking a view common in the finite element analysis, we interpret

the first N terms of the usual Fourier series solution as the exact

solution of an approximating problem in a subspace spanned by the

eigenfunctions of the underlying Sturm Liouville problem. This view

leads to a consistent solution technique for the heat, wave and

Poisson's equation, and allows an analysis of the error caused by

truncating the Fourier series. Applications to a variety of problems

will be discussed to demonstrate that the analytic approach remains a

valuable complement to purely numerical methods.

The talk is intended for students with an interest in actually

solving partial differential equations. It assumes a standard

background in undergraduate mathematics but not necessarily prior

exposure to the subject. The goal is to show that there is more to

separation of variables than is apparent from standard texts on

engineering mathematics.