a periodic boundary condition
Imagine bending the metal bar around into a ring so that its two ends meet and fuse — or studying heat flow around a circular loop, or a wave running around the rim of a circular pool. There is now no real 'edge': leaving the right end means re-entering at the left. A periodic boundary condition encodes exactly this wrap-around: it demands that the solution and its relevant derivatives match up where the two ends are identified, so the geometry has no boundary at all, only a seam that must be invisible.
Concretely, on an interval 0 <= x <= L that is wrapped into a circle of circumference L, the periodic conditions are u(0, t) = u(L, t) and u_x(0, t) = u_x(L, t): the value and the slope agree at the joined ends, so the solution continues smoothly through the seam as a function of period L. The natural building blocks are no longer pure sines or cosines but the full set of period-L modes — sin(2 pi n x / L) and cos(2 pi n x / L) together, or equivalently the complex exponentials e^(i 2 pi n x / L). This is precisely the setting of the ordinary Fourier series on a circle, which is why periodic conditions and Fourier analysis are so tightly linked.
Periodic conditions are the right choice whenever the physical domain truly closes on itself — a ring, a torus, an annular reactor, or an idealized infinite crystal whose unit cell repeats — and they are a favourite in computation because they remove troublesome edges and let you use the fast Fourier transform. One caveat worth flagging: like the pure Neumann problem, a periodic problem for Laplace's or Poisson's equation has its own solvability constraint and pins the solution only up to a constant, since a constant satisfies the equation and respects the seam. Periodicity removes the boundary but not every subtlety that boundaries used to settle.
Heat diffusing around a thin circular ring of circumference L obeys u_t = k u_xx with u(0, t) = u(L, t) and u_x(0, t) = u_x(L, t). Expand any initial profile in the period-L Fourier series; each mode e^(i 2 pi n x / L) decays as e^(-k (2 pi n / L)^2 t), so the ring smooths out to its average temperature.
Periodic = the two ends are glued; value and slope must match across the seam.
Periodic conditions are two conditions, not one: matching the value alone is not enough — the slope (and, for higher-order equations, further derivatives) must also agree, or the solution would have a kink at the seam.