Chemistry - Neumann boundary condition

(Redirected from Neumann problem)

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

$\frac{d^2y}{dx^2} + 3 y = 1$

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 : $\Omega\subset R^n$ such as

Δ y + y = 0

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

$\nu\cdot\nabla y(x) = f(x) \quad \forall x \in \partial\Omega$

Here, $ν$ denotes the (interior or exterior) normal to $\partial\Omega$ and $f$ is a given function.