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Difference between revisions of "Dirichlet boundary conditions"

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''Dirichlet conditions, Dirichlet data, boundary conditions of the first kind''
 
''Dirichlet conditions, Dirichlet data, boundary conditions of the first kind''
  
Consider a second-order partial differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110200/d1102001.png" /> on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110200/d1102002.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110200/d1102003.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110200/d1102004.png" /> (cf. also [[Differential equation, partial, of the second order|Differential equation, partial, of the second order]]). Boundary conditions of the form
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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 form
  
<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/d/d110/d110200/d1102005.png" /></td> </tr></table>
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$$u(x)=\phi(x),\quad x\in S,$$
  
 
are called Dirichlet boundary conditions.
 
are called Dirichlet boundary conditions.

Latest revision as of 14:55, 16 October 2014

Dirichlet conditions, Dirichlet data, boundary conditions of the first 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 form

$$u(x)=\phi(x),\quad x\in S,$$

are called Dirichlet boundary conditions.

A boundary value problem with Dirichlet conditions is also called a boundary value problem of the first kind (see First boundary value problem).

See also Second boundary value problem; Neumann boundary conditions; Third boundary value problem.

How to Cite This Entry:
Dirichlet boundary conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_boundary_conditions&oldid=33678
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article