In mathematics, a Neumann boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values the derivative of a solution is to take on the boundary of the domain.

In the case of an ordinary differential equation such as

on the interval [0,1] the Neumann boundary conditions take the form

y'(0) = 1

y'(1) = 2

For a partial differential equation on a domain : such as

Δy + y = 0

(Δ denotes the laplacian), the Neumann boundary condition typically takes the form

Here the vector-valued function f is specified.