the Neumann boundary condition
/ Neumann -> NOY-mahn /
Now imagine the rim of the metal plate is not held at a fixed temperature but wrapped in perfect insulation, so no heat can leak out across the edge. You are no longer saying what u is on the boundary; you are saying how u changes as you cross the boundary — specifically, that the outward flow there is a prescribed amount (zero, for perfect insulation). A Neumann boundary condition prescribes the normal derivative of u on the boundary: the rate at which u changes in the direction straight out of the region. Where Dirichlet fixes the value, Neumann fixes the slope across the edge.
Concretely, write the outward normal derivative as the derivative of u in the direction of the outward normal n, often denoted u_n or du/dn = grad u dotted with n. A Neumann condition reads u_n = g on the boundary, with g given. The physical reading is flux: by Fourier's law, heat flux is proportional to minus the temperature gradient, so prescribing u_n prescribes how much heat (or how much of whatever u measures) crosses the edge. The homogeneous case u_n = 0 means an insulated, no-flux, sealed wall — nothing enters or leaves. For the bar this is u_x(0, t) = 0 and u_x(L, t) = 0, and it leads naturally to cosine modes rather than sines.
Neumann problems carry a famous gotcha for steady states. For the pure Neumann problem with Laplace's or Poisson's equation, you cannot prescribe the boundary flux freely: integrating the equation forces a balance — for Laplacian u = 0 the total outward flux must be zero, and for Laplacian u = F the boundary flux must equal the total source inside. This is the Neumann compatibility condition. A second feature: a pure Neumann solution is only determined up to an added constant, because the conditions never mention the actual value of u, only its slopes — physically, an insulated bar has a temperature profile fixed only up to an overall offset.
Insulate both ends of the bar: u_x(0, t) = 0 and u_x(L, t) = 0. The eigenfunctions are now cosines, cos(n pi x / L), including the flat mode n = 0. Heat redistributes but none escapes, so the total heat is conserved and the bar settles to a uniform temperature equal to the average of its initial profile.
Neumann = fix the normal derivative (flux); homogeneous Neumann gives cosine eigenfunctions and conserves total heat.
A pure Neumann problem for Laplace/Poisson is not freely posable: the prescribed boundary flux must balance the interior sources (the Neumann compatibility condition), and even then the solution is unique only up to an additive constant.