Heterojunction

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

Auxx+2Buxy+Cuyy+Dux+Euy+F=0

that satisfies the condition

B2AC<0.

(Assuming implicitly that uxy=uyx. )

Just as one classifies conic sections and quadratic forms based on the discriminant B24AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B2AC, due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

Ax2+2Bxy+Cy2+=0 , which becomes (for : uxy=uyx=0) :
Auxx+Cuyy+Dux+Euy+F=0 , and Ax2+Cy2+=0 . This resembles the standard ellipse equation: x2a2+y2b21=0.

In general, if there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form

Lu=i=1nj=1nai,j2uxixj + (lower-order terms)=0, where L is an elliptic operator.

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

a2ux2+b2uxy+c2uy2+d2uyz+e2uz2 + (lower-order terms)=0,

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

a2ux2+c2uy2+e2uz2 + (lower-order terms)=0.

This can be compared to the equation for an ellipsoid; x2a2+y2b2+z2c2=1.

See also

External links

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