Editing Dirichlet boundary condition (section)

==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.