In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet (1805–1859). 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 .
Many other boundary conditions are possible. For example, there is the Cauchy boundary condition, or the mixed boundary condition which is a combination of the Dirichlet and Neumann conditions.
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