Heterojunction

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

$A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+D{u}_x+E{u}_y+F=0$

that satisfies the condition

$B^2-AC<0.$

(Assuming implicitly that $u_{xy}=u_{yx}$. )

Just as one classifies conic sections and quadratic forms based on the discriminant $B^2-4AC$, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by $B^2-AC,$ due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

$A{x}^2+2Bxy+C{y}^2+\cdots =0$ , which becomes (for : $u_{xy}=u_{yx}=0$) :

$A{u}_{xx}+C{u}_{yy}+D{u}_x+E{u}_y+F=0$ , and $A{x}^2+C{y}^2+\cdots =0$ . This resembles the standard ellipse equation: $\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0.$

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

$Lu=\sum _{i=1}^n\sum _{j=1}^n{a}_{i,j}\frac{{\partial }^{2}u}{\partial x_i\partial x_j}\text{ + (lower-order terms)}=0$, where L is an elliptic operator.

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

$a\frac{{\partial }^{2}u}{\partial x^2}+b\frac{{\partial }^{2}u}{\partial x\partial y}+c\frac{{\partial }^{2}u}{\partial y^2}+d\frac{{\partial }^{2}u}{\partial y\partial z}+e\frac{{\partial }^{2}u}{\partial z^2}\text{ + (lower-order terms)}=0,$

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

$a\frac{{\partial }^{2}u}{\partial x^2}+c\frac{{\partial }^{2}u}{\partial y^2}+e\frac{{\partial }^{2}u}{\partial z^2}\text{ + (lower-order terms)}=0.$

This can be compared to the equation for an ellipsoid; $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.$

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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• Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
  
  my web-site http://himerka.com/