Dirichlet boundary condition

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 along the boundary of the domain.

The question of finding solutions to such equations is known as the Dirichlet problem. In applied sciences, a Dirichlet boundary condition may also be referred to as a fixed boundary condition.

Examples

ODE

For an ordinary differential equation, for instance,

$y''+y=0$

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

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

where α and β are given numbers.

PDE

For a partial differential equation, for example,

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

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

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

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

Applications

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

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.