When solving the heat equation, $$ \partial_t u -\Delta u = f \text{ on } \Omega $$ what physical situations are represented by the following boundary conditions (on $\partial \Omega$)?

  • $u=g$ (Dirichlet condition),
  • $n\cdot\nabla u = h$ (Neumann condition),
  • $n\cdot\nabla u = \alpha u$ (Robin condition),
  • $n\cdot\nabla u = u^4-u_0^4$ (Stefan-Boltzmann condition).

Are there other common physical situations where another boundary condition is appropriate?


1 Answer 1


Different boundary conditions represent different models of cooling.

  • The first one states that you have a constant temperature at the boundary.This can be considered as a model of an ideal cooler in a good contact having infinitely large thermal conductivity

  • The second one states that we have a constant heat flux at the boundary. If the flux is equal zero, the boundary conditions describe the ideal heat insulator with the heat diffusion.

  • Robin boundary conditions are the mathematical formulation of the Newton's law of cooling where the heat transfer coefficient $\alpha$ is utilized. The heat transfer coefficient is determined by details of the interface structure (sharpness, geometry) between two media. This law describes quite well the boundary between metals and gas and is good for the convective heat transfer. http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html

  • The last one reflects the Stefan-Boltzman law and is good for describing the heat transfer due to radiation in vacuum

  • 1
    $\begingroup$ In the first case $g$ doesn't have to be constant, but the physical interpretation doesn't change (system couples to an external system which is kept at a fixed, position-dependent temperature). $\endgroup$
    – Vibert
    Commented May 28, 2013 at 21:23
  • $\begingroup$ @Vibert, yes, I think you are right. In general case, g could be some function, but I can’t find a practical example of applying of such boundary conditions. $\endgroup$
    – freude
    Commented May 29, 2013 at 11:11
  • 1
    $\begingroup$ @freude If you heat the workpiece from one side, and cool it from another, and have heating and cooling vary with time, then $g=g(x,t)$. $\endgroup$ Commented May 29, 2013 at 15:36
  • $\begingroup$ ok, that is fine ) $\endgroup$
    – freude
    Commented May 29, 2013 at 17:26

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