Difference between revisions of "Thermal-conductance equation"
From Encyclopedia of Mathematics
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''heat equation'' | ''heat equation'' | ||
The homogeneous partial differential equation | The homogeneous partial differential equation | ||
| − | + | $$\frac{\partial u}{\partial t}-a^2\sum_{k=1}^n\frac{\partial^2u}{\partial x_k^2}=0.$$ | |
| − | This equation is the simplest example of a [[Parabolic partial differential equation|parabolic partial differential equation]]. For | + | This equation is the simplest example of a [[Parabolic partial differential equation|parabolic partial differential equation]]. For $n=3$ it describes the process of heat diffusion in a solid body. The first boundary value problem (in a cylindrical domain) and the Cauchy–Dirichlet problem (in a half-space) are the fundamental well-posed problems for the thermal-conductance equation. A solution to the characteristic (Cauchy) problem can be given in explicit form: |
| − | + | $$u(x,t)=\frac{1}{(2a\sqrt{\pi t})^n}\int\limits_{\mathbf R^n}\exp\left(-\frac{|x-\xi|^2}{4a^2t}\right)\phi(\xi)d\xi,\quad t>0,$$ | |
| − | where | + | where $\phi(\xi)$ is a fixed continuous uniformly bounded function on $\mathbf R^n$. |
====References==== | ====References==== | ||
Revision as of 19:05, 30 July 2014
heat equation
The homogeneous partial differential equation
$$\frac{\partial u}{\partial t}-a^2\sum_{k=1}^n\frac{\partial^2u}{\partial x_k^2}=0.$$
This equation is the simplest example of a parabolic partial differential equation. For $n=3$ it describes the process of heat diffusion in a solid body. The first boundary value problem (in a cylindrical domain) and the Cauchy–Dirichlet problem (in a half-space) are the fundamental well-posed problems for the thermal-conductance equation. A solution to the characteristic (Cauchy) problem can be given in explicit form:
$$u(x,t)=\frac{1}{(2a\sqrt{\pi t})^n}\int\limits_{\mathbf R^n}\exp\left(-\frac{|x-\xi|^2}{4a^2t}\right)\phi(\xi)d\xi,\quad t>0,$$
where $\phi(\xi)$ is a fixed continuous uniformly bounded function on $\mathbf R^n$.
References
| [1] | A.V. Bitsadze, "The equations of mathematical physics" , MIR (1980) (Translated from Russian) |
| [2] | V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) |
Comments
References
| [a1] | J.R. Cannon, "The one-dimensional heat equation" , Addison-Wesley (1984) |
| [a2] | H.S. Carslaw, J.C. Jaeger, "Conduction of heat in solids" , Clarendon Press (1945) |
| [a3] | J. Cranck, "The mathematics of diffusion" , Clarendon Press (1975) |
| [a4] | A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) |
| [a5] | M. Jakob, "Heat transfer" , 1–2 , Wiley (1975) |
| [a6] | M.N. Ozisik, "Basic heat transfer" , McGraw-Hill (1977) |
| [a7] | D.V. Widder, "The heat equation" , Acad. Press (1975) |
How to Cite This Entry:
Thermal-conductance equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thermal-conductance_equation&oldid=32607
Thermal-conductance equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thermal-conductance_equation&oldid=32607
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article