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The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. ==Statement of the equation== For a function ''u''(''x'',''y'',''z'',''t'') of three spatial variables (''x'',''y'',''z'') (see cartesian coordinates) and the time variable ''t'', the heat equation is : More generally in any coordinate system: where ''α'' is a positive constant, and Δ or ∇2 denotes the Laplace operator. In the physical problem of temperature variation, ''u''(''x'',''y'',''z'',''t'') is the temperature and ''α'' is the thermal diffusivity. For the mathematical treatment it is sufficient to consider the case ''α'' = 1. Note that the state equation, given by the first law of thermodynamics (i.e. conservation of energy), is written in the following form (assuming no mass transfer, source or radiation). This form is more general and particular useful to recognise which property (e.g. ''cp'' or '''') influences which term. The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of Brownian motion via the Fokker–Planck equation. In financial mathematics it is used to solve the Black–Scholes partial differential equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes. 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 equation」の詳細全文を読む スポンサード リンク
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