---
tag: 3FW4J270FV
kind: concept
name: Periodic boundary conditions
msc: [34B05, 34B24]
sources:
  - url: https://en.wikipedia.org/wiki/Periodic_boundary_conditions
---

## Summary

Periodic boundary conditions ask a solution on an interval $[a, b]$ to take the same value and the same derivatives at both ends.
They turn a [[boundary value problem|TM8VQJWGHZ]] on an interval into a problem on a circle.
They appear in the study of equations with periodic coefficients and in Fourier series.
The eigenfunctions of $-y''$ with these conditions are the sines and cosines.

## Explanations

### L1 human

Imagine bending the interval $[a, b]$ into a ring so that its two ends meet.
A function that fits smoothly around the ring must have the same value and the same slope at both ends.
That is what periodic boundary conditions demand.
For example, on $[0, 2\pi]$ the function $\cos x$ satisfies them, since $\cos 0 = \cos 2\pi$ and $-\sin 0 = -\sin 2\pi$.
The function $x$ does not, since $0 \ne 2\pi$.