The partial differential equation
\frac{\partial^2u}{\partial t^2}= \frac{\partial^2 u}{\partial x^2}
and its extension to 2 or 3 space dimensions, is called "hyperbolic" in an obvious analogy with the hyperbolic equation
x^2- y^2= 1
In addition, just as the hyperbola has two asymptotes, so the hyperbolic differential equation has two "characteristic" lines that can be used to solve the equation.
The heat (or diffusion) equation,
\frac{\partial u}{\partial t}= \frac{\partial u^2}{\partial x^2}
has only a single characteristic line and is a "parabolic" equation.
Laplace's equation,
\frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0
has no characteristic lines and is an "elliptic" equation.
