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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 Initial value problem ： ウィキペディア英語版 Initial value problem In mathematics, in the field of differential equations, an initial value problem (also called the Cauchy problem by some authors) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. In physics or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential equation is an evolution equation specifying how, given initial conditions, the system will evolve with time. == Definition == An initial value problem is a differential equation :$y\text{'}(t)\; =\; f(t,\; y(t))$ with $f\backslash colon\; \backslash Omega\; \backslash subset\; \backslash mathbb\; \backslash times\; \backslash mathbb^n\; \backslash to\; \backslash mathbb^n$ where $\backslash Omega$ is an open set of $\backslash mathbb\; \backslash times\; \backslash mathbb^n$, together with a point in the domain of $f$ :$(t\_0,\; y\_0)\; \backslash in\; \backslash Omega$, called the initial condition. A solution to an initial value problem is a function $y$ that is a solution to the differential equation and satisfies :$y(t\_0)\; =\; y\_0$. In higher dimensions, the differential equation is replaced with a family of equations $y\_i\text{'}(t)=f\_i(t,\; y\_1(t),\; y\_2(t),\; \backslash dotsc)$, and $y(t)$ is viewed as the vector $(y\_1(t),\; \backslash dotsc,\; y\_n(t))$. More generally, the unknown function $y$ can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. $y\text{'}\text{'}(t)=f(t,y(t),y\text{'}(t))$. 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「Initial 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.