3 Classification of Second-order Equations

The three great equations of mathematical physics, the wave, heat and Laplace equations, behave in completely different ways. Waves travel at finite speed and keep their sharp edges, heat spreads instantly and smooths everything, and harmonic functions are determined everywhere by their boundary values. This section shows that every linear second-order equation in two variables is, locally, like one of the three, according to the sign of a single discriminant, and that the type decides which data make a problem well posed.

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.