Dirichlet boundary condition

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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.

In Finite Element Method, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.[2] The dependent unknown u in the same form as the weight function w appearing in the boundary expression is termed a primary variable, and its specification constitutes the essential or Dirichlet boundary condition.

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 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.


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

Other boundary conditions[edit]

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 also[edit]


  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.
  2. ^ J. N. Reddy, SECOND-ORDER DIFFERENTIAL EQUATIONS IN ONE DIMENSION: FINITE ELEMENT MODELS, An Introduction to the Finite Element Method, 3rd Edition, pp. 110