Neumann boundary condition

In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.[1] When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative applied at the boundary of the domain.

It is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed to its derivative) on the boundary, whereas the Cauchy boundary condition, mixed boundary condition and Robin boundary condition are all different types of combinations of the Neumann and Dirichlet boundary conditions.

Examples

ODE

For an ordinary differential equation, for instance,

${\displaystyle y''+y=0,}$

the Neumann boundary conditions on the interval [a,b] take the form

${\displaystyle y'(a)=\alpha ,\quad y'(b)=\beta ,}$

where α and β are given numbers.

PDE

For a partial differential equation, for instance,

${\displaystyle \nabla ^{2}y+y=0,}$

where 2 denotes the Laplace operator, the Neumann boundary conditions on a domain Ω ⊂ Rn take the form

${\displaystyle {\frac {\partial y}{\partial \mathbf {n} }}(\mathbf {x} )=f(\mathbf {x} )\quad \forall \mathbf {x} \in \partial \Omega ,}$

where n denotes the (typically exterior) normal to the boundary ∂Ω, and f is a given scalar function.

The normal derivative, which shows up on the left side, is defined as

${\displaystyle {\frac {\partial y}{\partial \mathbf {n} }}(\mathbf {x} )=\nabla y(\mathbf {x} )\cdot \mathbf {\hat {n}} (\mathbf {x} ),}$

where y(x) represents the gradient vector of y(x), is the unit normal, and represents the inner product operator.

It becomes clear that the boundary must be sufficiently smooth such that the normal derivative can exist, since, for example, at corner points on the boundary the normal vector is not well defined.

Applications

The following applications involve the use of Neumann boundary conditions: