Neumann boundary condition

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In mathematics, a Neumann boundary condition (named after Carl Neumann) 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

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

y'(0) = α₁

y'(1) = α₂

where α₁ and α₂ are given numbers.

For a partial differential equation on a domain

for example

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

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

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

[edit] See also

• Dirichlet boundary condition