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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 Biharmonic equation ： ウィキペディア英語版 Biharmonic equation In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. It is written as :$\backslash nabla^4\backslash varphi=0$ or :$\backslash nabla^2\backslash nabla^2\backslash varphi=0$ or :$\backslash Delta^2\backslash varphi=0$ where $\backslash nabla^4$ is the fourth power of the del operator and the square of the laplacian operator $\backslash nabla^2$ (or $\backslash Delta$), and it is known as the biharmonic operator or the bilaplacian operator. In summation notation, it can be written in $n$ dimensions as: : $$ \nabla^4\varphi=\sum_^n\sum_^n\partial_i\partial_i\partial_j\partial_j \varphi. For example, in three dimensional cartesian coordinates the biharmonic equation has the form : $$ + + + 2+ 2+ 2 = 0. As another example, in ''n''-dimensional Euclidean space, :$\backslash nabla^4\; \backslash left(\backslash right)=$ where :$r=\backslash sqrt.$ which, for ''n=3 and n=5'' only, becomes the biharmonic equation. A solution to the biharmonic equation is called a biharmonic function. Any harmonic function is biharmonic, but the converse is not always true. In two-dimensional polar coordinates, the biharmonic equation is :$$ \frac \frac \left(r \frac \left(\frac \frac \left(r \frac\right)\right)\right) + \frac \frac + \frac \frac - \frac \frac + \frac \frac = 0 which can be solved by separation of variables. The result is the Michell solution. ==2-dimensional space== The general solution to the 2-dimensional case is :$$ x v(x,y) - y u(x,y) + w(x,y) where $u(x,y)$, $v(x,y)$ and $w(x,y)$ are harmonic functions and $v(x,y)$ is a harmonic conjugate of $u(x,y)$. Just as harmonic functions in 2 variables are closely related to complex analytic functions, so are biharmonic functions in 2 variables. The general form of a biharmonic function in 2 variables can also be written as :$$ \operatorname(\barf(z) + g(z)) where $f(z)$ and $g(z)$ are analytic functions. 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「Biharmonic 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.