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 along the boundary of the domain.

In Finite Element Method, *essential* or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.^{[2]} The dependent unknown *u in the same form as the weight function w* appearing in the boundary expression is termed a *primary variable*, and its specification constitutes the *essential* or Dirichlet boundary condition.

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 [*a*,*b*] take the form

where α and β are given numbers.

### PDE

For a partial differential equation, for example,

where ∇^{2} denotes the Laplace operator, the Dirichlet boundary conditions on a domain *Ω* ⊂ ℝ^{n} take the form

where f is a known function defined on the boundary ∂*Ω*.

### Applications

For example, the following would be considered Dirichlet boundary conditions:

- In mechanical engineering and civil 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

## 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.**^**J. N. Reddy, SECOND-ORDER DIFFERENTIAL EQUATIONS IN ONE DIMENSION: FINITE ELEMENT MODELS,*An Introduction to the Finite Element Method*, 3rd Edition, pp. 110