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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 boundary value problem ： ウィキペディア英語版 Elliptic boundary value problem In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distribution of heat in a room several hours after the heating is turned on. Differential equations describe a large class of natural phenomena, from the heat equation describing the evolution of heat in (for instance) a metal plate, to the Navier-Stokes equation describing the movement of fluids, including Einstein's equations describing the physical universe in a relativistic way. Although all these equations are boundary value problems, they are further subdivided into categories. This is necessary because each category must be analyzed using different techniques. The present article deals with the category of boundary value problems known as linear elliptic problems. Boundary value problems and partial differential equations specify relations between two or more quantities. For instance, in the heat equation, the rate of change of temperature at a point is related to the difference of temperature between that point and the nearby points so that, over time, the heat flows from hotter points to cooler points. Boundary value problems can involve space, time and other quantities such as temperature, velocity, pressure, magnetic field, etc... Some problems do not involve time. For instance, if one hangs a clothesline between the house and a tree, then in the absence of wind, the clothesline will not move and will adopt a gentle hanging curved shape known as the catenary.〔Swetz, Faauvel, Bekken, "Learn from the Masters", 1997, MAA ISBN 0-88385-703-0, pp.128-9〕 This curved shape can be computed as the solution of a differential equation relating position, tension, angle and gravity, but since the shape does not change over time, there is no time variable. Elliptic boundary value problems are a class of problems which do not involve the time variable, and instead only depend on space variables. It is not possible to discuss elliptic boundary value problems in more detail without referring to calculus in multiple variables. Unless otherwise noted, all facts presented in this article can be found in.〔Partial Differential Equations by Lawrence C. Evans. American Mathematical Society, Providence, RI, 1998. Graduate Studies in Mathematics 19.〕 == The main example == In two dimensions, let $x,y$ be the coordinates. We will use the notation $u\_x,\; u\_$ for the first and second partial derivatives of $u$ with respect to $x$, and a similar notation for $y$. We will use the symbols $D\_x$ and $D\_y$ for the partial differential operators in $x$ and $y$. The second partial derivatives will be denoted $D\_x^2$ and $D\_y^2$. We also define the gradient $\backslash nabla\; u\; =\; (u\_x,u\_y)$, the Laplace operator $\backslash Delta\; u\; =\; u\_+u\_$ and the divergence $\backslash nabla\; \backslash cdot\; (u,v)\; =\; u\_x\; +\; v\_y$. Note from the definitions that $\backslash Delta\; u\; =\; \backslash nabla\; \backslash cdot\; (\backslash nabla\; u)$. The main example for boundary value problems is the Laplace operator, :$\backslash Delta\; u\; =\; f\; \backslash text\backslash Omega,$ :$u\; =\; 0\; \backslash text\; \backslash partial\; \backslash Omega;$ where $\backslash Omega$ is a region in the plane and $\backslash partial\; \backslash Omega$ is the boundary of that region. The function $f$ is known data and the solution $u$ is what must be computed. This example has the same essential properties as all other elliptic boundary value problems. The solution $u$ can be interpreted as the stationary or limit distribution of heat in a metal plate shaped like $\backslash Omega$, if this metal plate has its boundary adjacent to ice (which is kept at zero degrees, thus the Dirichlet boundary condition.) The function $f$ represents the intensity of heat generation at each point in the plate (perhaps there is an electric heater resting on the metal plate, pumping heat into the plate at rate $f(x)$, which does not vary over time, but may be nonuniform in space on the metal plate.) After waiting for a long time, the temperature distribution in the metal plate will approach $u$. 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 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.