# 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**.

## 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:

- 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

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

- Neumann boundary condition
- Mixed boundary condition
- Robin boundary condition
- Cauchy boundary condition
- Different types of boundary conditions in fluid dynamics

## References

- ↑ Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method,
*Engineering Analysis with Boundary Elements*,**29**, 268–302.