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. 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 take the form where α and β are given numbers. For a partial differential equation, for example, where ∇2 denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ ℝn take the form where f is a known function defined on the boundary ∂Ω.