# Dirichlet boundary condition

In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). 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:

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

For a partial differential equation, for instance:

$\nabla ^{2}y+y=0$ $y(x)=f(x)\quad \forall x\in \partial \Omega$ ### 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.