Neumann boundary condition
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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
- <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.