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).[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:

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

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

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

PDE

For a partial differential equation, for instance:

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

where ${\displaystyle \nabla ^{2}}$ denotes the Laplacian, 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 ${\displaystyle \partial \Omega }$.

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.