# Elliptic partial differential equation

An **elliptic partial differential equation** is a general partial differential equation of second order of the form

that satisfies the condition

Just as one classifies conic sections and quadratic forms based on the discriminant , the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

In general, if there are *n* independent variables *x*_{1}, *x*_{2 }, ..., *x*_{n}, a general linear partial differential equation of second order has the form

- , where L is an elliptic operator.

For example, in three dimensions (x,y,z) :

which, for completely separable u (i.e. u(x,y,z)=u(x)u(y)u(z) ) gives

This can be compared to the equation for an ellipsoid;

## See also

- Elliptic operator
- Hyperbolic partial differential equation
- Parabolic partial differential equation
- PDEs of second order, for fuller discussion

## External links

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