[1] When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain.
If Ω is the domain on which the given equation is to be solved and ∂Ω denotes its boundary, the Robin boundary condition is:[3] 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.
In one dimension, if, for example, Ω = [0,1], the Robin boundary condition becomes the conditions: 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.
Robin boundary conditions are commonly used in solving Sturm–Liouville problems which appear in many contexts in science and engineering.