Neumann boundary condition

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. In engineering applications, the following would be considered Neumann boundary conditions:

In thermodynamics, where a surface has a prescribed heat flux, such as a perfect insulator (where flux is zero) or an electrical component dissipating a known power.

For an ordinary differential equation, for instance:

y'' + y = 0~

the Neumann boundary conditions on the interval \( [a, \, b] \) take the form:

\( y'(a)= \alpha \ \text{and} \ y'(b) = \beta \)

where \alpha and \beta are given numbers.

For a partial differential equation, for instance:

\( \nabla^2 y + y = 0 \)

where \( \nabla^2 \) denotes the Laplacian, the Neumann boundary conditions on a domain \( \Omega \subset \mathbb{R}^n \) take the form:

\( \frac{\partial y}{\partial \mathbf{n}}(\mathbf{x}) = f(\mathbf{x}) \quad \forall \mathbf{x} \in \partial \Omega. \)

where \mathbf{n} denotes the (typically exterior) normal to the boundary \( \partial \Omega and f is a given scalar function.

The normal derivative which shows up on the left-hand side is defined as:

\( \frac{\partial y}{\partial \mathbf{n}}(\mathbf{x})=\nabla y(\mathbf{x})\cdot \mathbf{n}(\mathbf{x}) \)

where\( \nabla \) is the gradient (vector) and the dot is the inner product.

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.

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.
See also

Dirichlet boundary condition
Mixed boundary condition
Cauchy boundary condition
Robin boundary condition

References

Cheng, A. H. -D.; Cheng, D. T. (2005). "Heritage and early history of the boundary element method". Engineering Analysis with Boundary Elements 29 (3): 268. doi:10.1016/j.enganabound.2004.12.001. edit

Mathematics Encyclopedia