---
tag: GS3KT2H645
kind: concept
name: Neumann boundary condition
msc: [34B05, 35J25]
sources:
  - url: https://en.wikipedia.org/wiki/Neumann_boundary_condition
same_as:
  wikidata: Q1149279
  wikipedia: Neumann boundary condition
---

## Summary

A Neumann boundary condition prescribes the derivative of the unknown function in the direction normal to the boundary.
For an ordinary differential equation on $[a, b]$, it fixes $y'(a)$ and $y'(b)$.
For a partial differential equation on a domain, it fixes the outward normal derivative $\frac{\partial u}{\partial \nu}$ on the boundary, which is a flux.
It is one of the three classical conditions in a [[boundary value problem|TM8VQJWGHZ]], with the [[Dirichlet|EETVA6QYX4]] and Robin conditions.
See the [[definition|REH63MEBNC]].

## Explanations

### L1 human

A Neumann condition says how steeply the solution leaves the boundary, not where it is.
For heat flow, it prescribes the heat flux through the boundary.
A perfectly insulated wall lets no heat through, which is the condition $\frac{\partial u}{\partial \nu} = 0$.
For $y'' = 0$ on $[0, 1]$ with $y'(0) = y'(1) = 0$, every constant is a solution, so the solution is not unique.