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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 Lagrange's identity (boundary value problem) ： ウィキペディア英語版 Lagrange's identity (boundary value problem) In the study of ordinary differential equations and their associated boundary value problems, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator. Lagrange's identity is fundamental in Sturm–Liouville theory. In more than one independent variable, Lagrange's identity is generalized by Green's second identity. ==Statement== In general terms, Lagrange's identity for any pair of functions ''u'' and ''v''  in function space ''C''2 (that is, twice differentiable) in ''n'' dimensions is:〔 〕 :$vL()-uL^$ *()=\nabla \cdot \boldsymbol M, \ where: :$M\_i\; =\; \backslash sum\_^n\; a\_\backslash left($ v \frac -u \frac \right ) + uv \left( b_i - \sum_^ \frac \right ), and :$\backslash nabla\; \backslash cdot\; \backslash boldsymbol\; M\; =\; \backslash sum\_^n\; \backslash frac\; M\_i,$ The operator ''L'' and its adjoint operator ''L'' * are given by: :$L()\; =\; \backslash sum\_^n\; a\_\; \backslash frac\; +\; \backslash sum\_^n\; b\_i\; \backslash frac\; +c\; u$ and :$L^$ *() = \sum_^n \frac - \sum_^n \frac + cv. \, If Lagrange's identity is integrated over a bounded region, then the divergence theorem can be used to form Green's second identity in the form: :$\backslash int\_\backslash Omega\; v\; L()\backslash \; d\backslash Omega\; =\; \backslash int\_\; u\; L^$ *()\ d\Omega +\int_S \boldsymbol \, dS, where ''S'' is the surface bounding the volume ''Ω'' and ''n'' is the unit outward normal to the surface ''S''. ===Ordinary differential equations=== Any second order ordinary differential equation of the form: :$a(x)\backslash frac\; +\; b(x)\backslash frac\; +c(x)y\; +\backslash lambda\; w(x)\; y\; =0,$ can be put in the form:〔 〕 :$\backslash frac\; \backslash left(\; p(x)\; \backslash frac\; \backslash right\; )\; +\backslash left(\; q(x)+\; \backslash lambda\; w(x)\; \backslash right)\; y(x)\; =\; 0.$ This general form motivates introduction of the Sturm–Liouville operator ''L'', defined as an operation upon a function ''f '' such that: :$L\; f\; =\; \backslash frac\; \backslash left(\; p(x)\; \backslash frac\; \backslash right)\; +\; q(x)\; f.$ It can be shown that for any ''u'' and ''v'' for which the various derivatives exist, Lagrange's identity for ordinary differential equations holds:〔 :$uLv\; -\; vLu\; =\; -\; \backslash frac\; \backslash left(p(x)\; \backslash left(v\backslash frac\; -u\; \backslash frac\; \backslash right\; )\; \backslash right).$ For ordinary differential equations defined in the interval (1 ), Lagrange's identity can be integrated to obtain an integral form (also known as Green's formula):〔 〕〔 〕〔 〕 :$\backslash int\_0^1\; \backslash \; dx\; \backslash \; (\; uLv-vLu)\; =\; \backslash left(\backslash frac\; -\; v\; \backslash frac\; \backslash right)\backslash right)\_0^1,$ where $\backslash \; p=P(x)$, $\backslash \; q=Q(x)$, $\backslash \; u=U(x)$ and $\backslash \; v=V(x)$ are functions of $\backslash \; x$. $\backslash \; u$ and $\backslash \; v$ having continuous second derivatives on the 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「Lagrange's identity (boundary value problem)」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.