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.
For an ordinary differential equation, for instance,
the Dirichlet boundary conditions on the interval [a,b] take the form
where ? and ? are given numbers.
For a partial differential equation, for example,
where ?2 denotes the Laplace operator, the Dirichlet boundary conditions on a domain ? ? Rn take the form
where f is a known function defined on the boundary ??.
For example, the following would be considered Dirichlet boundary conditions: