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 a solution needs to take on the boundary of the domain. The question of finding solutions to such equations is known as the Dirichlet problem.
- 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 .
- Neumann boundary condition
- Mixed boundary condition
- Robin boundary condition
- Cauchy boundary condition
- Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements, 29, 268–302.