Dirichlet boundary condition
In mathematics, the Dirichlet boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet. 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.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 Laplace operator, the Dirichlet boundary conditions on a domain take the form
where is a known function defined on the boundary.Applications
For example, the following would be considered Dirichlet 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.
