View source for Thermal-conductance equation

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''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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,   "The equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.R. Cannon,   "The one-dimensional heat equation" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S. Carslaw,   J.C. Jaeger,   "Conduction of heat in solids" , Clarendon Press  (1945)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Cranck,   "The mathematics of diffusion" , Clarendon Press  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Jakob,   "Heat transfer" , '''1–2''' , Wiley  (1975)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.N. Ozisik,   "Basic heat transfer" , McGraw-Hill  (1977)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.V. Widder,   "The heat equation" , Acad. Press  (1975)</TD></TR></table>