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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 Heat kernel ： ウィキペディア英語版 Heat kernel In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time ''t'' = 0. The most well-known heat kernel is the heat kernel of ''d''-dimensional Euclidean space R''d'', which has the form of a time-varying Gaussian function, :$K(t,x,y)\; =\; \backslash frac\backslash ,$ This solves the heat equation :$\backslash frac(t,x,y)\; =\; \backslash Delta\_x\; K(t,x,y)\backslash ,$ for all ''t'' > 0 and ''x'',''y'' ∈ R''d'', with the initial condition :$\backslash lim\_\; K(t,x,y)\; =\; \backslash delta(x-y)=\backslash delta\_x(y)$ where δ is a Dirac delta distribution and the limit is taken in the sense of distributions. To wit, for every smooth function φ of compact support, :$\backslash lim\_\backslash int\_\; K(t,x,y)\backslash phi(y)\backslash ,dy\; =\; \backslash phi(x).$ On a more general domain Ω in R''d'', such an explicit formula is not generally possible. The next simplest cases of a disc or square involve, respectively, Bessel functions and Jacobi theta functions. Nevertheless, the heat kernel (for, say, the Dirichlet problem) still exists and is smooth for ''t'' > 0 on arbitrary domains and indeed on any Riemannian manifold with boundary, provided the boundary is sufficiently regular. More precisely, in these more general domains, the heat kernel for the Dirichlet problem is the solution of the initial boundary value problem :$\backslash frac(t,x,y)\; =\; \backslash Delta\; K(t,x,y)\; \backslash rm\; t>0\; \backslash rm\; x,y\backslash in\backslash Omega$ :$\backslash lim\_\; K(t,x,y)\; =\; \backslash delta\_x(y)\backslash rm\; x,y\backslash in\backslash Omega$ :$K(t,x,y)\; =\; 0,\; \backslash quad\; x\backslash in\backslash partial\backslash Omega\; \backslash rm\; y\backslash in\backslash partial\backslash Omega.$ It is not difficult to derive a formal expression for the heat kernel on an arbitrary domain. Consider the Dirichlet problem in a connected domain (or manifold with boundary) ''U''. Let λ''n'' be the eigenvalues for the Dirichlet problem of the Laplacian :$\backslash left\backslash$ \Delta \phi + \lambda \phi = 0 & \mathrm\ U\\ \phi=0 & \mathrm\ \partial U. \end\right. Let φ''n'' denote the associated eigenfunctions, normalized to be orthonormal in L2(''U''). The inverse Dirichlet Laplacian Δ−1 is a compact and selfadjoint operator, and so the spectral theorem implies that the eigenvalues satisfy : The heat kernel has the following expression: Formally differentiating the series under the sign of the summation shows that this should satisfy the heat equation. However, convergence and regularity of the series are quite delicate. The heat kernel is also sometimes identified with the associated integral transform, defined for compactly supported smooth φ by :$T\backslash phi\; =\; \backslash int\_\backslash Omega\; K(t,x,y)\backslash phi(y)\backslash ,dy.$ The spectral mapping theorem gives a representation of ''T'' in the form :$T\; =\; e^.$ There are several geometric results on heat kernels on manifolds; say, short-time asymptotics, long-time asymptotics, and upper/lower bounds of Gaussian type. ==See also== *Heat kernel signature *Minakshisundaram–Pleijel zeta function *Mehler kernel 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「Heat kernel」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.