mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Neumann (or second-type) boundary condition is a type of

boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...

, named after

Carl Neumann Carl Gottfried Neumann (also Karl; 7 May 1832 – 27 March 1925) was a German mathematician. Biography Neumann was born in Königsberg, Prussia, as the son of the mineralogist, physicist and mathematician Franz Ernst Neumann (1798–1895), who ...

. When imposed on an ordinary or a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...

, the condition specifies the values of the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

applied at the boundary of the

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...

. It is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition 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 function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...

, 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 In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...

, 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 of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

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, often ...

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, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...

, 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 A magnetic field is a vector 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 to its own velocity and to ...

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. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...

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 ...

for the magnetic scalar potential, the boundary condition is a Neumann condition. * In

spatial ecology Spatial ecology studies the ultimate distributional or spatial unit occupied by a species. In a particular habitat shared by several species, each of the species is usually confined to its own microhabitat or spatial niche because two species in ...

, a Neumann boundary condition on a

reaction–diffusion system Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the s ...

, such as

Fisher's equation In mathematics, Fisher's equation (named after statistician and biologist Ronald Fisher) also known as the Kolmogorov–Petrovsky–Piskunov equation (named after Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov), KPP equation or Fis ...

, 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