mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

, the condition specifies the values of the

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

applied at the boundary of the domain. It is possible to describe the problem using other boundary conditions: a

Dirichlet boundary condition In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the boundary of the domain are fixed. The question of finding solutions to such equat ...

specifies the values of the solution itself (as opposed to its derivative) on the boundary, whereas the Cauchy boundary condition, mixed boundary condition and Robin boundary condition are all different types of combinations of the Neumann and Dirichlet boundary conditions.

Examples

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 In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...

, the Neumann boundary conditions on a domain take the form :

\frac(\mathbf) = f(\mathbf) \quad \forall \mathbf \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(\mathbf) = \nabla y(\mathbf) \cdot \mathbf(\mathbf),

where represents the

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

vector of , is the unit normal, and represents the

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

, 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 Magnetostatics is the study of magnetic fields in systems where the electric currents, currents are steady current, steady (not changing with time). It is the magnetic analogue of electrostatics, where the electric charge, charges are stationary ...

, the

magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...

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 In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delt ...

Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...

for the

magnetic scalar potential Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electric potential. It is used to specify the magnetic H-field in cases when there are no free currents, in a manner analogous to using the electric ...

, 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 .

References

{{DEFAULTSORT:Neumann Boundary Condition Boundary conditions

Examples

ODE

PDE

Applications

See also

References