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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Elliptic partial differential equation ： ウィキペディア英語版 Elliptic partial differential equation An elliptic partial differential equation is a general partial differential equation of second order of the form : $Au\_\; +\; 2Bu\_\; +\; Cu\_\; +\; Du\_x\; +\; Eu\_y\; +\; F\; =\; 0\backslash ,$ that satisfies the condition : (Assuming implicitly that $u\_=u\_$. ) 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: : $Ax^2\; +\; 2Bxy\; +\; Cy^2\; +\; \backslash cdots\; =\; 0$ , which becomes (for : $u\_=u\_=0$) : : $Au\_\; +\; Cu\_\; +\; Du\_x\; +\; Eu\_y\; +\; F\; =\; 0$ , and $Ax^2\; +\; Cy^2\; +\; \backslash cdots\; =\; 0$ . This resembles the standard ellipse equation: $+-1=0.$ In general, if there are ''n'' independent variables ''x''1, ''x''2 , ..., ''x''''n'', a general linear partial differential equation of second order has the form : $L\; u\; =\backslash sum\_^n\backslash sum\_^n\; a\_\; \backslash frac\; \backslash quad\; \backslash text\; =0\; \backslash ,$, where L is an elliptic operator. For example, in three dimensions (x,y,z) : :$a\backslash frac\; +\; b\backslash frac\; +\; c\backslash frac\; +\; d\backslash frac\; +\; e\backslash frac\; \backslash text=\; 0,$ which, for completely separable u (i.e. u(x,y,z)=u(x)u(y)u(z) ) gives :$a\backslash frac\; +\; c\backslash frac\; +\; e\backslash frac\; \backslash text=\; 0.$ This can be compared to the equation for an ellipsoid; $++=1.$ == See also == * Elliptic operator * Hyperbolic partial differential equation * Parabolic partial differential equation *PDEs of second order, for fuller discussion 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Elliptic partial differential equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.