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In [[mathematics]], the '''Dirichlet''' (or '''first-type''') '''boundary condition''' is a type of [[boundary condition]], named after [[Peter Gustav Lejeune Dirichlet]] (1805–1859).<ref>Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method, ''Engineering Analysis with Boundary Elements'', '''29''', 268–302.</ref> When imposed on an [[ordinary differential equation|ordinary]] or a [[partial differential equation]], it specifies the values that a solution needs to take on the [[boundary (topology)|boundary]] of the domain.
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The question of finding solutions to such equations is known as the [[Dirichlet problem]]. In engineering applications, a Dirichlet boundary condition may also be referred to as a '''fixed boundary condition'''.
 
==Examples==
 
===ODE===
For an [[ordinary differential equation]], for instance:
 
:<math>y'' + y = 0~</math>
 
the Dirichlet boundary conditions on the interval <math>[a, \, b]</math> take the form:
 
:<math>y(a)= \alpha \ \text{and} \ y(b) = \beta</math>
 
where <math>\alpha</math> and <math>\beta</math> are given numbers.
 
===PDE===
For a [[partial differential equation]], for instance:
 
:<math>\nabla^2 y + y = 0</math>
 
where <math>\nabla^2</math> denotes the [[Laplace operator|Laplacian]], the Dirichlet boundary conditions on a domain <math>\Omega \subset \mathbb{R}^n</math> take the form:
 
:<math>y(x) = f(x) \quad \forall x \in \partial\Omega</math>
 
where ''f'' is a known [[function (mathematics)|function]] defined on the boundary <math>\partial\Omega</math>.
 
===Engineering applications===
For example, the following would be considered Dirichlet boundary conditions:
* In [[mechanical engineering]] ([[Euler–Bernoulli beam theory#Boundary considerations|beam theory]]), where one end of a beam is held at a fixed position in space.
* In [[thermodynamics]], 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]]
*[[Mixed boundary condition]]
*[[Robin boundary condition]]
*[[Cauchy boundary condition]]
*[[Different Types of Boundary Conditions in Fluid Dynamics]]
 
==References==
 
<references />
 
{{DEFAULTSORT:Dirichlet Boundary Condition}}
[[Category:Boundary conditions]]

Latest revision as of 06:26, 9 December 2014

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