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where a is a given function of x, y, z, t. Each is to hold for all t and for x = (x, y, z)
belonging to boundary of D. Usually, we write (D), (N), and (R) as equations. For
instance, (N) is written as the equation

∂u
= g(x, t) (5.3)
∂n

where g is a given function which could be called the boundary datum. Any of these
boundary conditions is called homogeneous if the specified function g(x, t) vanishes (equals
zero). Otherwise, it is called inhomogenous. As usual, n = (n 1 , n 2 , n 3 ) denotes the unit
normal vector on boundary of D, which points out-ward from D (see Figure 5.1). Also,
∂u/∂n ≡ n · ∇u denotes the directional derivative of u in the outward normal direction.

~n

D ~n
~n

~n

Figure 5.1: Domain D with unit normal vector on its boundary

In multidimensional problems the derivative of a function to each of the variables forms
a vector field (i.e., a function that takes a vector value at each point of space), usually
called the gradient. For three variables this takes the form

∂f ∂f ∂f
grad f(x, y, z) = ∇f(x, y, z) = (x, y, z), (x, y, z), (x, y, z)
∂x ∂y ∂z

normal

gradient

boundary

Figure 5.2: A sketch of the normal derivatives used in the von Neumann boundary
conditions

Typically we cannot specify the gradient at the boundary, since that is too restrictive
to allow for solutions. We can — and in physical problems often need to — specify the
component normal to the boundary, see Fig. 5.2 for an example. When this normal
derivative is specified we speak of von Neumann boundary conditions.
In one-dimensional problems where D is just an interval 0 < x < l, the boundary
consists of just the two endpoints, and these boundary conditions take the simple form

(D) u(0, t) = g(t) and u(l, t) = h(t)

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