Neumann boundary condition Information & Neumann boundary condition Links at HealthHaven.com

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:

$\frac{dy}{dx}(0) = \alpha_1$

$\frac{dy}{dx}(1) = \alpha_2$

where α₁ and α₂ are given numbers.

For a partial differential equation on a domain Ω⊂ℝⁿ such as:

$\nabla^2 y = 0$

where ∇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 ∂Ω and f is a given scalar function. The normal derivative which shows up on the left-hand side is defined as :

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

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

[edit] See also

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