Editing Dirichlet boundary condition (section)

{{short description|Type of constraint on solutions to differential equations}}
{{Differential equations |expanded=General topics}}
In mathematics, the '''Dirichlet''' '''boundary condition''' is imposed on an [[ordinary differential equation|ordinary]] or [[partial differential equation]], such that the values that the solution takes along the [[boundary (topology)|boundary]] of the domain are fixed. The question of finding solutions to such equations is known as the [[Dirichlet problem]]. In the sciences and engineering, a Dirichlet boundary condition may also be referred to as a '''fixed boundary condition''' or '''boundary condition of the first type'''. It is named after [[Peter Gustav Lejeune Dirichlet]] (1805–1859).<ref>{{cite journal |last=Cheng |first=A. |last2=Cheng |first2=D. T. |year=2005 |title=Heritage and early history of the boundary element method |journal=Engineering Analysis with Boundary Elements |volume=29 |issue=3 |pages=268–302 |doi=10.1016/j.enganabound.2004.12.001}}</ref>

In [[finite-element analysis]], the ''essential'' or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.<ref>{{cite book |first=J. N. |last=Reddy |authorlink=J. N. Reddy (engineer) |chapter=Second order differential equations in one dimension: Finite element models |title=An Introduction to the Finite Element Method |location=Boston |publisher=McGraw-Hill |year=2009 |edition=3rd |page=110 |isbn=978-0-07-126761-8 }}</ref> 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.