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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 Parabolic partial differential equation ： ウィキペディア英語版 Parabolic partial differential equation A parabolic partial differential equation is a type of second-order partial differential equation (PDE) of the form : $Au\_\; +\; 2Bu\_\; +\; Cu\_\; +\; Du\_x\; +\; Eu\_y\; +\; F\; =\; 0\backslash ,$ that satisfies the condition :$B^2\; -\; AC\; =\; 0.\backslash$ This definition is analogous to the definition of a planar parabola. This form of partial differential equation is used to describe a wide family of problems in science including heat diffusion, ocean acoustic propagation, physical or mathematical systems with a time variable, and processes that behave essentially like heat diffusing through a solid. A simple example of a parabolic PDE is the one-dimensional heat equation, :$u\_t\; =\; k\; u\_,\backslash$ where $u(t,x)$ is the temperature at time $t$ and at position $x$, and $k$ is a constant. The symbol $u\_t$ signifies the partial derivative with respect to the time variable $t$, and similarly $u\_$ is the second partial derivative with respect to $x$. This equation says, roughly, that temperature at a given time and point rises or falls at a rate proportional to the difference between the temperature at that point and the average temperature near that point. The quantity $u\_$ measures how far off the temperature is from satisfying the mean value property of harmonic functions. A generalization of the heat equation is :$u\_t\; =\; -Lu,\backslash$ where $L$ is a second-order elliptic operator (implying $L$ must be positive also; a case where $L$ is non-positive is described below). Such a system can be hidden in an equation of the form :$\backslash nabla\; \backslash cdot\; (a(x)\; \backslash nabla\; u(x))\; +\; b(x)^T\; \backslash nabla\; u(x)\; +\; cu(x)\; =\; f(x)$ if the matrix-valued function $a(x)$ has a kernel of dimension 1. == Solution == Under broad assumptions, parabolic PDEs as given above have solutions for all ''x,y'' and ''t''>0. An equation of the form $u\_t\; =\; -L(u)$ is considered parabolic if ''L'' is a (possibly nonlinear) function of ''u'' and its first and second derivatives, with some further conditions on ''L''. With such a nonlinear parabolic differential equation, solutions exist for a short time—but may explode in a singularity in a finite amount of time. Hence, the difficulty is in determining solutions for all time, or more generally studying the singularities that arise. This is in general quite difficult, as in the solution of the Poincaré conjecture via Ricci flow. 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「Parabolic partial differential 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.