ODE For an ordinary differential equation, for instance, :y'' + y = 0, the Neumann boundary conditions on the interval take the form :y'(a)= \alpha, \quad y'(b) = \beta, where and are given numbers.
PDE For a partial differential equation, for instance, :\nabla^2 y + y = 0, where denotes the
Laplace operator, the Neumann boundary conditions on a domain take the form :\frac{\partial y}{\partial \mathbf{n}}(\mathbf{x}) = f(\mathbf{x}) \quad \forall \mathbf{x} \in \partial \Omega, where denotes the (typically exterior)
normal to the
boundary , and is a given
scalar function. The
normal derivative, which shows up on the left side, is defined as :\frac{\partial y}{\partial \mathbf{n}}(\mathbf{x}) = \nabla y(\mathbf{x}) \cdot \mathbf{\hat{n}}(\mathbf{x}), where represents the
gradient vector of , is the unit normal, and represents the
inner product operator. It becomes clear that the boundary must be sufficiently smooth such that the normal derivative can exist, since, for example, at corner points on the boundary the normal vector is not well defined.
Applications The following applications involve the use of Neumann boundary conditions: • In
thermodynamics, a prescribed heat flux from a surface would serve as boundary condition. For example, a perfect insulator would have no flux while an electrical component may be dissipating at a known power. • In
magnetostatics, the
magnetic field intensity can be prescribed as a boundary condition in order to find the
magnetic flux density distribution in a magnet array in space, for example in a permanent magnet motor. Since the problems in magnetostatics involve solving
Laplace's equation or
Poisson's equation for the
magnetic scalar potential, the boundary condition is a Neumann condition. • In
spatial ecology, a Neumann boundary condition on a
reaction–diffusion system, such as
Fisher's equation, can be interpreted as a reflecting boundary, such that all individuals encountering are reflected back onto . ==See also==