Dirichlet boundary condition

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In the mathematical study of differential equations, 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 along the boundary of the domain.

In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.[2] The dependent unknown u in the same form as the weight function w appearing in the boundary expression is termed a primary variable, and its specification constitutes the essential or Dirichlet boundary condition.

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.

Examples[edit]

ODE[edit]

For an ordinary differential equation, for instance,

the Dirichlet boundary conditions on the interval [a,b] take the form
where α and β are given numbers.

PDE[edit]

For a partial differential equation, for example,

where 2 denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ Rn take the form
where f is a known function defined on the boundary ∂Ω.

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.; Cheng, D. T. (2005). "Heritage and early history of the boundary element method". Engineering Analysis with Boundary Elements. 29 (3): 268–302. doi:10.1016/j.enganabound.2004.12.001.
  2. ^ Reddy, J. N. (2009). "Second order differential equations in one dimension: Finite element models". An Introduction to the Finite Element Method (3rd ed.). Boston: McGraw-Hill. p. 110. ISBN 978-0-07-126761-8.