An elliptic partial differential equation is a general partial differential equation of second order
that satisfies the condition
(Assuming implicitly that . )
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:
- , which becomes (for : ) :
- , and . This resembles the standard ellipse equation:
In general, if there are n independent variables x1, x2, ..., xn, 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;
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