Hyperbolic partial differential equation

A hyperbolic partial differential equation is usually a second-order partial differential equation of the form

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:A u_{xx} + 2 B u_{xy} + C u_{yy} + D u_x + E u_y + F = 0

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

with det egin{pmatrix} A & B \ B & C end{pmatrix} = A C - B^2 < 0.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The wave equation:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:rac{partial^2 u}{partial t^2} - abla^2 u = 0

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

is such a hyperbolic equation.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~