翻訳と辞書 Words near each other ・ Duisburger Musikpreis ・ Duisburg–Dortmund railway ・ Duisburg–Quakenbrück railway ・ Duisenberg School of Finance ・ Duiske Abbey ・ Duisky ・ Duissern (VRR) ・ Duhallow ADS Commonwealth War Graves Commission Cemetery ・ Duhallow GAA ・ Duhallow Junior A Football Championship ・ Duhallow Junior A Hurling Championship ・ Duhamel ・ Duhamel Broadcasting Tower Angora ・ Duhamel municipal election, 2009 ・ Duhamel's integral ・ Duhamel's principle ・ Duhamel, Alberta ・ Duhamel, Quebec ・ Duhamel-Ouest ・ Duhaney Park F.C. ・ Duhat ・ Duhat, Laguna ・ Duhat, Philippines ・ Duhat, Quezon ・ Duhemia ・ Duhem–Margules equation ・ Duhem–Quine thesis ・ Duhla Peak ・ Duhnen ・ Duhok International Airport Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Duhamel's principle ： ウィキペディア英語版 Duhamel's principle In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the heat equation, wave equation, and vibrating plate equation. It is named after Jean-Marie Duhamel who first applied the principle to the inhomogeneous heat equation that models, for instance, the distribution of heat in a thin plate which is heated from beneath. For linear evolution equations without spatial dependency, such as a harmonic oscillator, Duhamel's principle reduces to the method of variation of parameters technique for solving linear inhomogeneous ordinary differential equations.〔Fritz John, "Partial Differential Equations', New York, Springer-Verlag, 1982, 4th ed., 0387906096〕 The philosophy underlying Duhamel's principle is that it is possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example of the heat equation modeling the distribution of heat energy ''u'' in R''n''. The initial value problem is :$$ \begin u_t(x,t) - \Delta u(x,t) = 0 &(x,t)\in \mathbf^n\times (0,\infty)\\ u(x,0) = g(x) & x\in \mathbf^n \end where ''g'' is the initial heat distribution. By contrast, the inhomogeneous problem for the heat equation is :$$ \begin u_t(x,t) -\Delta u(x,t) = f(x,t) &(x,t)\in \mathbf^n\times (0,\infty)\\ u(x,0) = 0 & x\in \mathbf^n \end corresponds to adding an external heat energy ''ƒ''(''x'',''t'')''dt'' at each point. Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice ''t'' = ''t''0. By linearity, one can add up (integrate) the resulting solutions through time ''t''0 and obtain the solution for the inhomogeneous problem. This is the essence of Duhamel's principle. ==General considerations== Formally, consider a linear inhomogeneous evolution equation for a function :$u:D\backslash times(0,\backslash infty)\backslash to\; \backslash mathbf$ with spatial domain ''D'' in R''n'', of the form :$$ \begin u_t(x,t) -Lu(x,t) = f(x,t) &(x,t)\in D\times (0,\infty)\\ u|_ = 0 &\\ u(x,0) = 0 & x\in D, \end where ''L'' is a linear differential operator that involves no time derivatives. Duhamel's principle is, formally, that the solution to this problem is :$u(x,t)\; =\; \backslash int\_0^t\; (P^sf)(x,t)\backslash ,ds$ where ''P''''s''''ƒ'' is the solution of the problem :$$ \begin u_t - Lu = 0 & (x,t)\in D\times (s,\infty)\\ u|_ = 0 &\\ u(x,s) = f(x,s) & x\in D. \end Duhamel's principle also holds for linear systems (with vector-valued functions ''u''), and this in turn furnishes a generalization to higher ''t'' derivatives, such as those appearing in the wave equation (see below). Validity of the principle depends on being able to solve the homogeneous problem in an appropriate function space and that the solution should exhibit reasonable dependence on parameters so that the integral is well-defined. Precise analytic conditions on ''u'' and ''f'' depend on the particular application. 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「Duhamel's principle」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.