Neumann boundary condition

Sometimes you do not control the value at the edge of a region but rather the flow across it. Wrap a rod in perfect insulation and you are not fixing the end temperatures — you are demanding that no heat crosses the ends. To fix a temperature you would name a value; to fix a flow you must name a slope, because flux is proportional to the gradient. The Neumann boundary condition is the prescription that fixes that slope — the normal derivative of the field on the boundary.

Formally, on the boundary you require partial u / partial n = h, where partial / partial n is the derivative in the direction normal (perpendicular) to the boundary and h is a given function. By Fourier's law and its analogues, that normal derivative is exactly the flux through the wall, so a Neumann condition specifies how much heat, charge, or fluid crosses the boundary. The most important special case is the homogeneous Neumann condition partial u / partial n = 0, which says no flux at all — a perfectly insulated wall, an impermeable barrier, or a free (unforced) edge. For the heat or Laplace equation on an interval, this condition makes the spatial eigenfunctions cosines rather than sines, and it admits a constant mode (the n = 0 term) that a Dirichlet condition would forbid.

Neumann conditions are the right model whenever a boundary is insulated, sealed, or symmetric: an insulated rod end, the surface of a thermally isolated body, a wall a fluid cannot pass through, or a plane of symmetry where the flux must vanish. A pure Neumann problem for Laplace's or Poisson's equation has a subtlety: the solution is determined only up to an additive constant (you can raise the whole temperature uniformly without changing any slope), and for Poisson's equation a solution exists only if the total source balances the total prescribed boundary flux — a built-in conservation law called the compatibility or solvability condition.