Dirichlet boundary condition

For an ordinary differential equation, for instance,

${\displaystyle y''+y=0}$

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

${\displaystyle y(a)=\alpha ,\quad y(b)=\beta ,}$

where α and β are given numbers.

For a partial differential equation, for example,

${\displaystyle \nabla ^{2}y+y=0,}$

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

${\displaystyle y(x)=f(x)\quad \forall x\in \partial \Omega ,}$

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

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

• In mechanical engineering and 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.