Neumann boundary condition
From Free net encyclopedia
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
- <math>
\frac{d^2y}{dx^2} + 3 y = 1 </math>
on the interval <math>[0,1],</math> the Neumann boundary condition takes the form
- <math>y'(0) = \alpha_1</math>
- <math>y'(1) = \alpha_2</math>
where <math>\alpha_1</math> and <math>\alpha_2</math> are given numbers.
For a partial differential equation on a domain
- <math>\Omega\subset R^n,</math>
for example
- <math>
\Delta y + y = 0 </math>
(<math>\Delta</math> denotes the Laplacian), the Neumann boundary condition takes the form
- <math>
\frac{\partial y}{\partial \nu}(x) = f(x) \quad \forall x \in \partial\Omega. </math>
Here, <math>\nu</math> denotes the (typically exterior) normal to the boundary ∂Ω and <math>f</math> is a given function. The normal derivative which shows up on the left-hand side is defined as
- <math>\frac{\partial y}{\partial \nu}(x)=\nabla y(x)\cdot \nu (x)</math>
where ∇ is the gradient and the dot is the inner product.