Dirichlet boundary condition

In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the 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).^[1]

In finite-element analysis, the 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.

^ 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.
^ 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.

[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.

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