Difference between revisions of "Neumann boundary conditions"
From Encyclopedia of Mathematics
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''Neumann conditions, Neumann data, boundary conditions of the second kind'' | ''Neumann conditions, Neumann data, boundary conditions of the second kind'' | ||
| − | Consider a second-order partial differential equation | + | Consider a second-order partial differential equation $Lu=f$ on a domain $D$ in $\mathbf R^n$ with boundary $S$ (cf. also [[Differential equation, partial, of the second order|Differential equation, partial, of the second order]]). Boundary conditions of the type |
| − | + | $$\frac{\partial u(x)}{\partial n}=\phi(x),\quad x\in S,$$ | |
| − | where | + | where $n$ is the outward pointing normal at $x$, are called Neumann boundary conditions. |
A boundary value problem with Neumann conditions is also called a boundary value problem of the second kind (see [[Second boundary value problem|Second boundary value problem]]). | A boundary value problem with Neumann conditions is also called a boundary value problem of the second kind (see [[Second boundary value problem|Second boundary value problem]]). | ||
See also [[First boundary value problem|First boundary value problem]]; [[Dirichlet boundary conditions|Dirichlet boundary conditions]]; [[Third boundary value problem|Third boundary value problem]]. | See also [[First boundary value problem|First boundary value problem]]; [[Dirichlet boundary conditions|Dirichlet boundary conditions]]; [[Third boundary value problem|Third boundary value problem]]. | ||
Latest revision as of 07:13, 23 August 2014
Neumann conditions, Neumann data, boundary conditions of the second kind
Consider a second-order partial differential equation $Lu=f$ on a domain $D$ in $\mathbf R^n$ with boundary $S$ (cf. also Differential equation, partial, of the second order). Boundary conditions of the type
$$\frac{\partial u(x)}{\partial n}=\phi(x),\quad x\in S,$$
where $n$ is the outward pointing normal at $x$, are called Neumann boundary conditions.
A boundary value problem with Neumann conditions is also called a boundary value problem of the second kind (see Second boundary value problem).
See also First boundary value problem; Dirichlet boundary conditions; Third boundary value problem.
How to Cite This Entry:
Neumann boundary conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_boundary_conditions&oldid=18697
Neumann boundary conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_boundary_conditions&oldid=18697
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article