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 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.

Examples

ODE

For an ordinary differential equation, for instance: the Dirichlet boundary conditions on the interval take the form: where and are given numbers.

PDE

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.