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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 Neumann boundary condition ： ウィキペディア英語版 Neumann boundary condition In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, it specifies the values that the derivative of a solution is to take on the boundary of the domain. In engineering applications, the following would be considered Neumann boundary conditions: * In thermodynamics, where a surface has a prescribed heat flux, such as a perfect insulator (where flux is zero) or an electrical component dissipating a known power. For an ordinary differential equation, for instance: :$y\text{'}\text{'}\; +\; y\; =\; 0~$ the Neumann boundary conditions on the interval $(\backslash ,\; b)$ take the form: :$y\text{'}(a)=\; \backslash alpha\; \backslash \; \backslash text\; \backslash \; y\text{'}(b)\; =\; \backslash beta$ where $\backslash alpha$ and $\backslash beta$ are given numbers. * For a partial differential equation, for instance: :$\backslash nabla^2\; y\; +\; y\; =\; 0$ where $\backslash nabla^2$ denotes the Laplacian, the Neumann boundary conditions on a domain $\backslash Omega\; \backslash subset\; \backslash mathbb^n$ take the form: :$\backslash frac)\; =\; f(\backslash mathbf)\; \backslash quad\; \backslash forall\; \backslash mathbf\; \backslash in\; \backslash partial\; \backslash Omega.$ where $\backslash mathbf$ denotes the (typically exterior) normal to the boundary $\backslash partial\; \backslash Omega$ and ''f'' is a given scalar function. The normal derivative which shows up on the left-hand side is defined as: :$\backslash frac)=\backslash nabla\; y(\backslash mathbf)\backslash cdot\; \backslash mathbf(\backslash mathbf)$ where $\backslash nabla$ is the gradient (vector) and the dot is the inner product. It becomes clear that the boundary must be sufficiently smooth such that the normal derivative can exist, since for example at corner points of the boundary the normal vector is not well defined. Many other boundary conditions are possible. For example, there is the Cauchy boundary condition, or the mixed boundary condition which is a combination of the Neumann and Dirichlet conditions. ==See also== *Dirichlet boundary condition *Mixed boundary condition *Cauchy boundary condition *Robin boundary condition 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「Neumann boundary condition」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.