# Dirichlet boundary condition

In the mathematical study of differential equations, 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 along the boundary of the domain.

In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.[2] The dependent unknown u in the same form as the weight function w appearing in the boundary expression is termed a primary variable, and its specification constitutes the essential or Dirichlet boundary condition.

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 [a,b] take the form
${\displaystyle y(a)=\alpha ,\quad y(b)=\beta ,}$
where α and β 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 Ω ⊂ Rn take the form
${\displaystyle y(x)=f(x)\quad \forall x\in \partial \Omega ,}$
where f is a known function defined on the boundary ∂Ω.

### 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.