Robin boundary condition

The Robin boundary condition specifies a linear combination of the value of a function and the value of its derivative at the boundary of a given domain. It is a generalization of the Dirichlet boundary condition, which specifies only the function's value, and the Neumann boundary condition, which specifies only the function's derivative. A common physical example is in heat transfer, where a surface might lose heat to the environment via convection. The rate of heat flow (related to the derivative of temperature) would be proportional to the difference between the surface temperature (the value of the temperature function) and the ambient temperature.

Other equivalent names in use are Fourier-type condition and radiation condition.

Definition

Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This contrasts to mixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary. Robin boundary conditions are also called impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems (Hahn, 2012).

If Ω is the domain on which the given equation is to be solved and ∂Ω denotes its boundary, the Robin boundary condition is:

${\displaystyle au+b\frac{\mathrm{\partial }u}{\mathrm{\partial }n}=g\text{on }\mathrm{\partial }\mathrm{\Omega }}$

for some non-zero constants a and b and a given function g defined on ∂Ω. Here, u is the unknown solution defined on Ω and ⁠∂u/∂n⁠ denotes the normal derivative at the boundary. More generally, a and b are allowed to be (given) functions, rather than constants.

In one dimension, if, for example, Ω = [0,1], the Robin boundary condition becomes the conditions:

${\displaystyle \begin{array}{cc}au(0)-b{u}^{\prime }(0)& =g(0)\\ au(1)+b{u}^{\prime }(1)& =g(1)\end{array}}$

Notice the change of sign in front of the term involving a derivative: that is because the normal to [0,1] at 0 points in the negative direction, while at 1 it points in the positive direction.