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An '''elliptic partial differential equation''' is a general [[partial differential equation]] of second [[Differential equation#Nomenclature|order]] of the form | |||
: <math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\,</math> | |||
that satisfies the condition | |||
:<math>B^2 - AC < 0.\ </math> | |||
(Assuming implicitly that <math>u_{xy}=u_{yx}</math>. ) | |||
Just as one classifies [[conic section]]s and [[quadratic form]]s based on the [[discriminant]] <math>B^2 - 4AC</math>, the same can be done for a second-order PDE at a given point. However, the [[discriminant]] in a PDE is given by <math>B^2 - AC,</math> due to the convention (discussion and explanation [[partial differential equations#Equations of second order|here]]). The above form is analogous to the equation for a planar [[ellipse]]: | |||
: <math>Ax^2 + 2Bxy + Cy^2 + \cdots = 0</math> , which becomes (for : <math>u_{xy}=u_{yx}=0</math>) : | |||
: <math>Au_{xx} + Cu_{yy} + Du_x + Eu_y + F = 0</math> , and <math>Ax^2 + Cy^2 + \cdots = 0</math> . This resembles the standard ellipse equation: <math>{x^2\over a^2}+{y^2\over b^2}-1=0.</math> | |||
In general, if there are ''n'' [[independent variable]]s ''x''<sub>1</sub>, ''x''<sub>2 </sub>, ..., ''x''<sub>''n''</sub>, a general linear partial differential equation of second order has the form | |||
: <math>L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\part^2 u}{\partial x_i \partial x_j} \quad \text{ + (lower-order terms)} =0 \,</math>, where L is an [[elliptic operator]]. | |||
For example, in three dimensions (x,y,z) : | |||
:<math>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,</math> | |||
which, for [[separable equation|completely separable]] u (i.e. u(x,y,z)=u(x)u(y)u(z) ) gives | |||
:<math>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.</math> | |||
This can be compared to the equation for an ellipsoid; <math>{x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1. </math> | |||
== See also == | |||
* [[Elliptic operator]] | |||
* [[Hyperbolic partial differential equation]] | |||
* [[Parabolic partial differential equation]] | |||
*[[partial differential equations#Equations of second order|PDEs of second order]], for fuller discussion | |||
==External links== | |||
* {{springer|title=Elliptic partial differential equation|id=p/e035520}} | |||
* {{springer|title=Elliptic partial differential equation, numerical methods|id=p/e035530}} | |||
[[Category:Partial differential equations]] |
Revision as of 21:47, 24 December 2013
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 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;
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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