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.
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 instance:
where denotes the Laplacian, the Dirichlet boundary conditions on a domain take the form:
where f is a known function defined on the boundary .
For example, the following would be considered Dirichlet boundary conditions:
- In mechanical engineering (beam theory), where one end of a beam is held at a fixed position in space.
- In thermodynamics, where a surface is held at a fixed temperature.
- In electrostatics, where a node of a circuit is held at a fixed voltage.
- In fluid dynamics, the no-slip condition for viscous fluids states that at a solid boundary, the fluid will have zero velocity relative to the boundary.
Other boundary conditions
- Neumann boundary condition
- Mixed boundary condition
- Robin boundary condition
- Cauchy boundary condition
- Different types of boundary conditions in fluid dynamics
- Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements, 29, 268–302.