Second-order PDEs of two variables are of the form
and can be classified on the basis of the discriminant, 
, into
three types
based upon the conic sections of the same name:
Examples of hyperbolic, parabolic, and elliptic equations are the classical wave equation, the diffusion equation, and Laplace's equation, representing the application of PDEs to sound and light, heat, and electrostatic phenomena, respectively. The type of solution for each of these equation types is quite different. Elliptic equations produce stationary and energy-minimizing solutions, parabolic equations produce a smooth-spreading flow of an initial disturbance, and hyperbolic equations produce a propagating disturbance. For example, the first PDE encountered by a student may be the one-dimensional wave equation,
since it predicts the time evolution of many types of transverse and
longitudinal waves.[3,4] Replacing
y in eq:GeneralPDE by t, we see that the classical wave
equation is an example of a hyperbolic PDE. Since a suitable
change of variables will reduce any hyperbolic equation to this canonical
form (to first order) we are really studying a very general problem.
Table 
lists the PDEs that can be solved
by the accompanying program, WAVE. As is typical in
computational work, units will be chosen to minimize
roundoff erros. Notice, for instance, that mass and Planck's
constant are unity in the Schrödinger equation.
Table: Partial Differential Equations solved by WAVE.
Another useful classification of partial differential equations is based
on a property called linearity. Assume two different solutions of
a differential equation, 
and 
, have been found.
Then the equation is said to be linear if their sum
is also a solution of the equation. Linearity is a very desirable
property in many systems, since it allows us to express the time
evolution of an arbitrary initial state, 
, in terms of
the time evolution of some complete set of known and presumably simple
functions. (See Chapter 2 for a discussion of Fourier methods.) For
example, if the sound of a violin and a piano are processed
simultaneously by a stereo system, then the listener should be able to
hear the resultant music as if each instrument were played separately
and the sound intensity added together. Any nonlinearity in the stereo system is
viewed as an undesirable distortion. Other phenomena, such as laser-pulse
compression or frequency doubling, depend on system nonlinearity.
Both linear and nonlinear systems will be studied in the exercises.
