Dirichlet boundary condition

Examples

ODE

For an ordinary differential equation, for instance, y* + y = 0, the Dirichlet boundary conditions on the interval [a,*b''] take the form y(a) = \alpha, \quad y(b) = \beta, where α and β are given numbers.

PDE

For a partial differential equation, for example, \nabla^2 y + y = 0, where \nabla^2 denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ Rn take the form y(x) = f(x) \quad \forall x \in \partial\Omega, 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 heat transfer, 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.

References

References

1. Cheng, A.. (2005). "Heritage and early history of the boundary element method". [[Engineering Analysis with Boundary Elements]].
2. Reddy, J. N.. (2009). "An Introduction to the Finite Element Method". McGraw-Hill.