Open main menu

In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859).[1] 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.

Contents

ExamplesEdit

ODEEdit

For an ordinary differential equation, for instance,

 

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

 

where α and β are given numbers.

PDEEdit

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.

ApplicationsEdit

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

Other boundary conditionsEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements, 29, 268–302.