Dirichlet boundary condition

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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[edit]

ODE[edit]

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

where \alpha and \beta are given numbers.

PDE[edit]

For a partial differential equation, for instance:

\nabla^2 y + y = 0

where \nabla^2 denotes the Laplacian, the Dirichlet boundary conditions on a domain \Omega \subset \mathbb{R}^n take the form:

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

where f is a known function defined on the boundary \partial\Omega.

Engineering applications[edit]

For example, the following would be considered Dirichlet boundary conditions:

Other boundary conditions[edit]

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.

See also[edit]

References[edit]

  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.