Dirichlet boundary condition
From Wikipedia, the free encyclopedia
In mathematics, a Dirichlet boundary condition (often referred to as a first-type boundary condition) imposed on an ordinary differential equation or a partial differential equation specifies the values a solution is to take on the boundary of the domain. The question of finding solutions to such equations is known as the Dirichlet problem.
In the case of an ordinary differential equation such as
on the interval [0,1] the Dirichlet boundary conditions take the form
- y(0) = α1
- y(1) = α2
where α1 and α2 are given numbers.
For a partial differential equation on a domain
such as
- Δy + y = 0
(Δ denotes the Laplacian), the Dirichlet boundary condition takes the form
where f is a known function defined on the boundary ∂Ω.
Dirichlet boundary conditions are perhaps the easiest to understand but there are many other conditions possible. For example, there is the Neumann boundary condition or the mixed boundary condition which is a combination of the Dirichlet and Neumann conditions.
