Neumann boundary condition

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, for example such as

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

on the interval $[0,1],$ the Neumann boundary condition takes the form

y'(0) = α 1

y'(1) = α 2

where $α 1$ and $α 2$ are given numbers.

For a partial differential equation on a domain

$\Omega\subset R^n,$

for example

Δ y + y = 0

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

$\frac{\partial y}{\partial \nu}(x) = f(x) \quad \forall x \in \partial\Omega.$

Here, $ν$ denotes the (typically exterior) normal to the boundary ∂Ω and $f$ is a given function. The normal derivative which shows up on the left-hand side is defined as

$\frac{\partial y}{\partial \nu}(x)=\nabla y(x)\cdot \nu (x)$

where ∇ is the gradient and the dot is the inner product.

Neumann boundary condition

See also

See also: Neumann boundary condition, Boundary (topology), Boundary condition, Dirichlet boundary condition, Gradient, Inner product, Laplace operator, Mathematics, Normal vector, Ordinary differential equation