Dirichlet boundary conditions

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

For an ordinary differential equation, for instance,

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

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

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