# Neumann boundary condition

In mathematics, a Neumann boundary condition 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

[itex]

\frac{d^2y}{dx^2} + 3 y = 1 [itex]

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

[itex]y'(0) = \alpha_1[itex]
[itex]y'(1) = \alpha_2[itex]

where [itex]\alpha_1[itex] and [itex]\alpha_2[itex] are given numbers.

For a partial differential equation on a domain

[itex]\Omega\subset R^n,[itex]

for example

[itex]

\Delta y + y = 0 [itex]

([itex]\Delta[itex] denotes the Laplacian), the Neumann boundary condition takes the form

[itex]

\frac{\partial y}{\partial \nu}(x) = f(x) \quad \forall x \in \partial\Omega. [itex]

Here, [itex]\nu[itex] denotes the (typically exterior) normal to the boundary ∂Ω and [itex]f[itex] is a given function. The normal derivative which shows up on the left-hand side is defined as

[itex]\frac{\partial y}{\partial \nu}(x)=\nabla y(x)\cdot \nu (x)[itex]

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

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