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:
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.
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.
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 Neumann and Dirichlet conditions.
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—Ralph Waldo Emerson (1803–1882)
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—Ambrose Bierce (1842–1914)