# Parabolic PDEs

The diffusion equation:

$$\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0;$$
is a parabolic equation since the discriminant is zero (B² - 4AC = 0² - (4)(0)(-1) = 0).

Here are some more well-known parabolic equations:

(1) Heat equation which could be used to model a thermodynamics problem with
transient heat transfer;
(2) Blasius boundary layer equation which describes the non-dimensional velocity
distribution in the laminar boundary layer over a flat plate.

Generally, parabolic equations have very smooth solutions.