# 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 applied sciences, a Dirichlet boundary condition may also be referred to as a fixed boundary condition.

## Examples

### ODE

For an ordinary differential equation, for instance,

${\displaystyle y''+y=0}$

the Dirichlet boundary conditions on the interval ${\displaystyle [a,b]}$ take the form

${\displaystyle y(a)=\alpha ,\quad y(b)=\beta ,}$

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are given numbers.

### PDE

For a partial differential equation, for example,

${\displaystyle \nabla ^{2}y+y=0,}$

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

${\displaystyle y(x)=f(x)\quad \forall x\in \partial \Omega ,}$

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

### 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.