# Dirichlet boundary condition

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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 that a solution needs to take on along the boundary of the domain.

The question of finding solutions to such equations is known as the Dirichlet problem. In engineering applications, a Dirichlet boundary condition may also be referred to as a fixed 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:

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

where $\alpha$ and $\beta$ are given numbers.

### PDE

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 $\partial\Omega$.

### Engineering applications

For example, the following would be considered Dirichlet boundary conditions:

## 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.