Neumann boundary condition
Encyclopedia
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
.
the Neumann boundary conditions on the interval
take the form:
where
and 
are given numbers.
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.
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 Dirichlet and Neumann conditions.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann
Carl Neumann
Carl Gottfried Neumann was a German mathematician.Neumann was born in Königsberg, Prussia, as the son of the mineralogist, physicist and mathematician Franz Ernst Neumann , who was professor of mineralogy and physics at Königsberg University...
.
When imposed on an ordinary
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
or a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
, it specifies the values that the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of a solution is to take on the boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of the domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
.
- 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 LaplacianLaplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
, the Neumann boundary conditions on a domain
take the form:where
denotes the (typically exterior) normal to the boundaryBoundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of 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 gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
(vector) and the dot is the inner product.
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 Dirichlet and Neumann conditions.
See also
- Dirichlet boundary conditionDirichlet boundary conditionIn mathematics, the Dirichlet boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet who studied under Cauchy and succeeded Gauss at University of Göttingen. When imposed on an ordinary or a partial differential equation, it specifies the values a...
- Mixed boundary conditionMixed boundary conditionIn mathematics, a mixed boundary condition for a partial differential equation indicates that different boundary conditions are used on different parts of the boundary of the domain of the equation....
- Cauchy boundary conditionCauchy boundary conditionIn mathematics, a Cauchy boundary condition imposed on an ordinary differential equation or a partial differential equation specifies both the values a solution of a differential equation is to take on the boundary of the domain and the normal derivative at the boundary. It corresponds to imposing...
- Robin boundary conditionRobin boundary conditionIn mathematics, the Robin boundary condition is a type of boundary condition, named after Victor Gustave Robin . When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the...