# Elliptic PDEs

Laplace's equation is a common example of the elliptic equation:

\begin{equation} \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0; \end{equation}

since the discriminant is negative (

*B² - 4AC = 0² - (4)(1)(1) = -4*).

Here are some more examples of the elliptic equations:

(1) Potential flow: flow around an airfoil (without separation);

(2) Deformation of a membrane damped over its edges.

Similar to the parabolic equations, this type of equations has also smooth solutions.

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