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).[1] When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take on along the boundary of the domain.

The question of finding solutions to such equations is known as the Dirichlet problem. In engineering applications, 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 and are given numbers.


For a partial differential equation, for example:

where denotes the Laplacian, the Dirichlet boundary conditions on a domain take the form:

where f is a known function defined on the boundary .

Engineering 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.

See also


  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.
This article is issued from Wikipedia - version of the 11/23/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.