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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 Robin boundary condition ： ウィキペディア英語版 Robin boundary condition In mathematics, the Robin boundary condition (; properly ), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897).〔Gustafson, K., (1998). Domain Decomposition, Operator Trigonometry, Robin Condition, ''Contemporary Mathematics'', 218. 432–437.〕 When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function ''and'' the values of its derivative on the boundary of the domain. Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This contrasts to mixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary. Robin boundary conditions are also called impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems (Hahn, 2012). If Ω is the domain on which the given equation is to be solved and $\backslash partial\backslash Omega$ denotes its boundary, the Robin boundary condition is: :$a\; u\; +\; b\; \backslash frac\; =g\; \backslash qquad\; \backslash text\; ~\; \backslash partial\; \backslash Omega\backslash ,$ for some non-zero constants ''a'' and ''b'' and a given function ''g'' defined on $\backslash partial\backslash Omega$. Here, ''u'' is the unknown solution defined on $\backslash Omega$ and $/$ denotes the normal derivative at the boundary. More generally, ''a'' and ''b'' are allowed to be (given) functions, rather than constants. In one dimension, if, for example, $\backslash Omega\; =\; ()$, the Robin boundary condition becomes the conditions: :$a\; u(0)\; -\; bu\text{'}(0)\; =g(0)\backslash ,$ :$a\; u(1)\; +\; bu\text{'}(1)\; =g(1).\backslash ,$ notice the change of sign in front of the term involving a derivative: that is because the normal to $()$ at 0 points in the negative direction, while at 1 it points in the positive direction. Robin boundary conditions are commonly used in solving Sturm–Liouville problems which appear in many contexts in science and engineering. In addition, the Robin boundary condition is a general form of the insulating boundary condition for convection–diffusion equations. Here, the convective and diffusive fluxes at the boundary sum to zero: :$u\_x(0)\backslash ,c(0)\; -D\; \backslash frac=0\backslash ,$ where ''D'' is the diffusive constant, ''u'' is the convective velocity at the boundary and ''c'' is the concentration. The second term is a result of Fick's law of diffusion. ==See also== *Dirichlet boundary condition *Neumann boundary condition *Mixed boundary condition *Cauchy 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「Robin 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.