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 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. 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.
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.
For a partial differential equation, for example,
where ∇2 denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ ℝn take the form
where f is a known function defined on the boundary ∂Ω.
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
- 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