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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