TABLE OF CONTENTS A wide variety of partial differential equations occurs in technical computing. We cannot begin to cover them all in this book. In this chapter, we limit ourselves to three model problems for second-order partial differential equations in one or two space dimensions.
All the problems we consider involve the Laplacian operator, which is
in one space dimension and
in two space dimensions. We let 
denote the single variable x in one dimension and the pair of variables ( x, y) in two dimensions.
The first model problem is the Poisson equation. This elliptic equation does not involve a time variable, and so describes the steady state, quiescent behavior of a model variable:
There are no initial conditions.
The second model problem is the heat equation. This parabolic equation occurs in models involving diffusion and decay:
The initial condition is
The third model problem is the wave equation. This hyperbolic equation describes how a disturbance travels through matter. If the units are chosen so that the wave propagation speed is equal to one, the amplitude of a wave satisfies
Typical initial conditions specify the initial amplitude and take the initial velocity to be zero:
In one dimension, all the problems take place on a finite interval on the x-axis. In more than one space dimension, geometry plays a vital role. In two dimensions, all the problems take place in a bounded region ? in the ( x, y) plane. In...
