Editing Dirichlet boundary condition

{{short description|Type of constraint on solutions to differential equations}}
{{Differential equations |expanded=General topics}}
In mathematics, the '''Dirichlet''' '''boundary condition''' is imposed on an [[ordinary differential equation|ordinary]] or [[partial differential equation]], such that the values that the solution takes along the [[boundary (topology)|boundary]] of the domain are fixed. The question of finding solutions to such equations is known as the [[Dirichlet problem]]. In the sciences and engineering, a Dirichlet boundary condition may also be referred to as a '''fixed boundary condition''' or '''boundary condition of the first type'''. It is named after [[Peter Gustav Lejeune Dirichlet]] (1805–1859).<ref>{{cite journal |last=Cheng |first=A. |last2=Cheng |first2=D. T. |year=2005 |title=Heritage and early history of the boundary element method |journal=Engineering Analysis with Boundary Elements |volume=29 |issue=3 |pages=268–302 |doi=10.1016/j.enganabound.2004.12.001}}</ref>

In [[finite-element analysis]], the ''essential'' or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.<ref>{{cite book |first=J. N. |last=Reddy |authorlink=J. N. Reddy (engineer) |chapter=Second order differential equations in one dimension: Finite element models |title=An Introduction to the Finite Element Method |location=Boston |publisher=McGraw-Hill |year=2009 |edition=3rd |page=110 |isbn=978-0-07-126761-8 }}</ref> The dependent unknown ''u in the same form as the weight function w'' appearing in the boundary expression is termed a ''primary variable'', and its specification constitutes the ''essential'' or Dirichlet boundary condition. 

==Examples==

===ODE===
For an [[ordinary differential equation]], for instance, <math display="block">y'' + y = 0,</math> the Dirichlet boundary conditions on the interval {{math|[''a'',''b'']}} take the form
<math display="block">y(a) = \alpha, \quad y(b) = \beta,</math>
where {{mvar|α}} and {{mvar|β}} are given numbers.

===PDE===

For a [[partial differential equation]], for example, <math display="block">\nabla^2 y + y = 0,</math> where <math>\nabla^2</math> denotes the [[Laplace operator]], the Dirichlet boundary conditions on a domain {{math|Ω ⊂ '''R'''<sup>''n''</sup>}} take the form
<math display="block">y(x) = f(x) \quad \forall x \in \partial\Omega,</math>
where {{mvar|f}} is a known [[function (mathematics)|function]] defined on the boundary {{math|∂Ω}}.

===Applications===

For example, the following would be considered Dirichlet boundary conditions:
* In [[mechanical engineering]] and [[civil engineering]] ([[Euler–Bernoulli beam theory#Boundary considerations|beam theory]]), where one end of a beam is held at a fixed position in space.
* In [[heat transfer]], where a surface is held at a fixed temperature.
* In [[electrostatics]], where a node of a circuit is held at a fixed voltage.
* In [[fluid dynamics]], the [[no-slip condition]] for viscous fluids states that at a solid boundary, the fluid will have zero velocity relative to the boundary.

==Other boundary conditions==

Many other boundary conditions are possible, including the [[Cauchy boundary condition]] and the [[mixed boundary condition]]. The latter is a combination of the Dirichlet and [[Neumann boundary condition|Neumann]] conditions.

==See also==
*[[Neumann boundary condition]]
*[[Robin boundary condition]]
*[[Boundary conditions in fluid dynamics]]

==References==

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{{DEFAULTSORT:Dirichlet Boundary Condition}}
[[Category:Boundary conditions]]