# Parabolic PDEs

The diffusion equation:

\begin{equation} \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0; \end{equation}

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.

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