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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 Separable partial differential equation ： ウィキペディア英語版 Separable partial differential equation A separable partial differential equation (PDE) is one that can be broken into a set of separate equations of lower dimensionality (fewer independent variables) by a method of separation of variables. This generally relies upon the problem having some special form or symmetry. In this way, the PDE can be solved by solving a set of simpler PDEs, or even ordinary differential equations (ODEs) if the problem can be broken down into one-dimensional equations. The most common form of separation of variables is simple separation of variables in which a solution is obtained by assuming a solution of the form given by a product of functions of each individual coordinate.There is a special form of separation of variables called $R$-separation of variables which is accomplished by writing the solution as a particular fixed function of the coordinates multiplied by a product of functions of each individual coordinate. Laplace's equation on $^n$ is an example of a partial differential equation which admits solutions through $R$-separation of variables; in the three-dimensional case this uses 6-sphere coordinates. (This should not be confused with the case of a separable ODE, which refers to a somewhat different class of problems that can be broken into a pair of integrals; see separation of variables.) == Example == For example, consider the time-independent Schrödinger equation :$(+\; V(\backslash mathbf))\backslash psi(\backslash mathbf)\; =\; E\backslash psi(\backslash mathbf)$ for the function $\backslash psi(\backslash mathbf)$ (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function $V(\backslash mathbf)$ in three dimensions is of the form :$V(x\_1,x\_2,x\_3)\; =\; V\_1(x\_1)\; +\; V\_2(x\_2)\; +\; V\_3(x\_3),$ then it turns out that the problem can be separated into three one-dimensional ODEs for functions $\backslash psi\_1(x\_1)$, $\backslash psi\_2(x\_2)$, and $\backslash psi\_3(x\_3)$, and the final solution can be written as $\backslash psi(\backslash mathbf)\; =\; \backslash psi\_1(x\_1)\; \backslash cdot\; \backslash psi\_2(x\_2)\; \backslash cdot\; \backslash psi\_3(x\_3)$. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.〔L. P. Eisenhart, "Enumeration of potentials for which one-particle Schrodinger equations are separable," ''Phys. Rev.'' 74, 87-89 (1948).〕) 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「Separable 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.