ODE For an
ordinary differential equation, for instance, y'' + y = 0, the Dirichlet boundary conditions on the interval take the form y(a) = \alpha, \quad y(b) = \beta, where and are given numbers.
PDE For a
partial differential equation, for example, \nabla^2 y + y = 0, where \nabla^2 denotes the
Laplace operator, the Dirichlet boundary conditions on a domain take the form y(x) = f(x) \quad \forall x \in \partial\Omega, where is a known
function defined on the boundary .
Applications 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
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==