Dirichlet boundary condition

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:

$y'' + y = 0~$

the Dirichlet boundary conditions on the interval $[a, \, b]$ take the form:

$y(a)= \alpha \ \text{and} \ y(b) = \beta$

where α and β are given numbers.

• For a partial differential equation, for instance:

$\nabla^2 y + y = 0$

where $\nabla^2$ denotes the Laplacian, the Dirichlet boundary conditions on a domain $\Omega \subset \mathbb{R}^n$ take the form:

$y(x) = f(x) \quad \forall x \in \partial\Omega$

where f is a known function defined on the boundary $\partial\Omega$.

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.

See also

• Neumann boundary condition
• Mixed boundary condition
• Robin boundary condition
• Cauchy boundary condition