Dirichlet boundary condition

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Not to be confused with Dirichlet conditions.

In mathematics, the Dirichlet (or first type) boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet (1805-1859). [1] When imposed on an ordinary or a partial differential equation, it specifies the values a solution needs 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

\frac{d^2y}{dx^2} + 3 y = 1

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

\Omega\subset R^n

such as

Δy + y = 0

(Δ denotes the Laplacian), the Dirichlet boundary condition takes the form

y(x) = f(x) \quad \forall x \in \partial\Omega

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.

[edit] See also

[edit] References

  1. ^ Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements, 29, 268–302.