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Difference between revisions of "Thermal-conductance equation"

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

Latest revision as of 06:08, 30 June 2024

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)
[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=55826
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article