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, it specifies the values that 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 conditions take the form:

$\begin{align} \frac{dy}{dx}(0) &= \alpha_1 \\ \frac{dy}{dx}(1) &= \alpha_2 \end{align}$

where α₁ and α₂ are given numbers.

For a partial differential equation on a domain $\Omega \subset \mathbb{R}^n$ such as:

$\nabla^2 y = 0$

where $\nabla^2$ denotes the Laplacian, the Neumann boundary condition takes the form:

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

Here, n denotes the (typically exterior) normal to the boundary $\partial\Omega$ and f is a given scalar function. The normal derivative which shows up on the left-hand side is defined as :

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

where $\nabla$ is the gradient (vector) and the dot is the inner product with the (unit) normal vector n.

See also

• Dirichlet boundary condition
• Mixed boundary condition
• Cauchy boundary condition
• Robin boundary condition

References

1. ^ Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements, 29, 268–302.