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∂u ∂u
(N) (0, t) = g(t) and (l, t) = h(t)
∂x ∂x
and similarly for the Robin condition.
Following are some illustrations of physical problems corresponding to these boundary
conditions.

5.1 The vibrating string

If the string is held fixed at both ends, as for a violin string, we have the homogeneous
Dirichlet conditions u(0, t) = u(l, t) = 0.
Imagine, on the other hand, that one end of the string is free to move transversally
without any resistance (say, along a frictionless track); then there is no tension T at that
end, so u x = 0. This is a Neumann condition.
Third, the Robin condition would be the correct one if one were to imagine that an
end of the string were free to move along a track but were attached to a coiled spring or
rubber band (obeying Hooke’s law) which tended to pull it back to equilibrium position.
In that case the string would exchange some of its energy with the coiled spring.
Finally, if an end of the string were simply moved in a specified way, we would have
an inhomogeneous Dirichlet condition at that end.

5.2 Diffusion

If the diffusing substance is enclosed in a container D so that none can escape or enter,
then the concentration gradient in the normal direction must vanish, by Fick’s law. Thus
∂u/∂n = 0 on S = boundary of D, which is the Neumann condition. If, on the other
hand, the container is permeable and is so constructed that any substance which escapes
to the boundary of the container is immediately washed away, then we have u = 0 on S.

5.3 Heat

Heat conduction is described by the diffusion equation with u(x, t) = temperature. If
the object D through which the heat is flowing is perfectly insulated, then no heat flows
across the boundary and we have the Neumann condition ∂u/∂n = 0.
On the other hand, if the object were immersed in a large reservoir of specified tem-
perature g(t) and there were perfect thermal conduction, then we’d have the Dirichlet
condition u = g(t) on boundary of D. Suppose that we had a uniform rod insulated along
its length 0 ≤ x ≤ l, whose end at x = l were immersed in the reservoir of temperature
g(t). If heat were exchanged between the end and the reservoir so as to obey Newton ’ s
law of cooling, then
∂u
(l, t) = −a(u(l, t) − g(t)),
∂x
where a > 0. Heat from the hot rod radiates into the cool reservoir. This is an inhomo-
geneous Robin condition.

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