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Difference between revisions of "Neumann boundary conditions"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110020/n1100201.png" /> on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110020/n1100202.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110020/n1100203.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110020/n1100204.png" /> (cf. also [[Differential equation, partial, of the second order|Differential equation, partial, of the second order]]). Boundary conditions of the type
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110020/n1100205.png" /></td> </tr></table>
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$$\frac{\partial u(x)}{\partial n}=\phi(x),\quad x\in S,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110020/n1100206.png" /> is the outward pointing normal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n110/n110020/n1100207.png" />, are called Neumann boundary conditions.
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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=33090
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article