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| An '''elliptic partial differential equation''' is a general [[partial differential equation]] of second [[Differential equation#Nomenclature|order]] of the form
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| : <math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\,</math>
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| that satisfies the condition
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| :<math>B^2 - AC < 0.\ </math>
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| (Assuming implicitly that <math>u_{xy}=u_{yx}</math>. )
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| 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]]:
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| : <math>Ax^2 + 2Bxy + Cy^2 + \cdots = 0</math> , which becomes (for : <math>u_{xy}=u_{yx}=0</math>) : | | Catrina Le is what's constructed on her birth diploma though she [http://www.bbc.Co.uk/search/?q=doesn%27t+essentially doesn't essentially] like being called prefer that. Software creating a is where her first income comes from so soon her husband but also her will start your own [http://Www.Dict.cc/?s=business business]. What the woman loves doing is to go to karaoke but this woman is thinking on starting something new. For years she's been living with Vermont. She is running and maintaining a blog here: http://prometeu.net<br><br>My site: how to hack clash of clans ([http://prometeu.net prometeu.net]) |
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| : <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>
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| 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
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| : <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]].
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| 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> | |
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| which, for [[separable equation|completely separable]] u (i.e. u(x,y,z)=u(x)u(y)u(z) ) gives
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| :<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>
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| 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>
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| == See also ==
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| * [[Elliptic operator]]
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| * [[Hyperbolic partial differential equation]]
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| * [[Parabolic partial differential equation]]
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| *[[partial differential equations#Equations of second order|PDEs of second order]], for fuller discussion
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| ==External links==
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| * {{springer|title=Elliptic partial differential equation|id=p/e035520}}
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| * {{springer|title=Elliptic partial differential equation, numerical methods|id=p/e035530}}
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| [[Category:Partial differential equations]]
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Catrina Le is what's constructed on her birth diploma though she doesn't essentially like being called prefer that. Software creating a is where her first income comes from so soon her husband but also her will start your own business. What the woman loves doing is to go to karaoke but this woman is thinking on starting something new. For years she's been living with Vermont. She is running and maintaining a blog here: http://prometeu.net
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