Neumann boundary condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. 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.
- For an ordinary differential equation, for instance:
the Neumann boundary conditions on the interval take the form:
where and are given numbers.
- For a partial differential equation, for instance:
where denotes the Laplacian, the Neumann boundary conditions on a domain take the form:
The normal derivative which shows up on the left-hand side is defined as:
It becomes clear that the boundary must be sufficiently smooth such that the normal derivative can exist, since for example at corner points of the boundary the normal vector is not well defined.
In problems of heat diffusion, it is common to encounter the condition that no heat may enter or leave the boundary of the domain, I.e. that the domain is perfectly insulated. This corresponds to the Neumann boundary where the normal derivative is zero.
See also 
- Dirichlet boundary condition
- Mixed boundary condition
- Cauchy boundary condition
- Robin boundary condition
- Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements, 29, 268–302.