Neumann boundary condition

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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 condition takes 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

\Omega\subset \mathbb R^n,

for example

\nabla^{2} y + y = 0

(\nabla^{2} 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 scalar 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.

[edit] See also

[edit] 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.