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Dirichlet boundary condition

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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). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take on along the 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.

ODE

For an ordinary differential equation, for instance,

y ″ + y = 0

the Dirichlet boundary conditions on the interval [ a , b ] take the form

y ( a ) = α , y ( b ) = β ,

where α and β are given numbers.

PDE

For a partial differential equation, for example,

∇ 2 y + y = 0 ,

where ∇ 2 denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ R n take the form

y ( x ) = f ( x ) ∀ x ∈ ∂ Ω ,

where f is a known function defined on the boundary ∂ Ω .

Engineering applications

For example, the following would be considered Dirichlet boundary conditions:

  • In mechanical engineering or civil engineering (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 conditions.

    References

    Dirichlet boundary condition Wikipedia
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