In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.[1] 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.
the Neumann boundary conditions on the interval take the form:
where and are given numbers.
where denotes the Laplacian, the Neumann boundary conditions on a domain take the form:
where denotes the (typically exterior) normal to the boundary 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 (vector) and the dot is the inner product.
Many other boundary conditions are possible. For example, there is the Cauchy boundary condition, or the mixed boundary condition which is a combination of the Dirichlet and Neumann conditions.