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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