13. Two-dimensional problems

You have taken your first step into a larger world.

—Obi-Wan Kenobi, Star Wars: A New Hope

We have graduated from ODEs, usually having either time or space as the independent variable, to PDEs in which both space and time are represented simultaneously. The final innovation in this book is to consider problems with two space dimensions. We will confine ourselves to the simplest possible class of two-dimensional regions, leaving aside the major issue of the geometry of the domain. Even so there are many important and valuable mathematical models within this constraint.

First we explore how to add another space dimension to the two classes of PDE encountered so far: the parabolic and hyperbolic equations. Last comes the third major class of PDE: elliptic equations. These problems omit time altogether and therefore are used to represent steady-state phenomena.