In mathematics, the Dirichlet boundary condition is imposed on an ordinary or

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

, 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 Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ...

(1805–1859). In finite-element analysis, the ''essential'' or Dirichlet boundary condition is defined by weighted-integral form of a differential equation. 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.

Examples

ODE

For an

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

, for instance,

y'' + y = 0,

the Dirichlet boundary conditions on the interval take the form

y(a) = \alpha, \quad y(b) = \beta,

where and are given numbers.

PDE

For a

, for example,

\nabla^2 y + y = 0,

where

\nabla^2

denotes the

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...

, the Dirichlet boundary conditions on a domain take the form

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

where is a known function defined on the boundary .

Applications

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

mechanical engineering Mechanical engineering is the study of physical machines and mechanism (engineering), mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and engineering mathematics, mathematics principl ...

and

civil engineering Civil engineering is a regulation and licensure in engineering, professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads ...

( beam theory), where one end of a beam is held at a fixed position in space. * In

heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...

, where a surface is held at a fixed temperature. * In

electrostatics Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...

, where a node of a circuit is held at a fixed voltage. * In

fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...

, the

no-slip condition In fluid dynamics, the no-slip condition is a Boundary conditions in fluid dynamics, boundary condition which enforces that at a solid boundary, a viscous fluid attains zero bulk velocity. This boundary condition was first proposed by Osborne Reyno ...

for viscous fluids states that at a solid boundary, the fluid will have zero velocity relative to the boundary.

Other boundary conditions

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.

References

{{DEFAULTSORT:Dirichlet Boundary Condition Boundary conditions

Examples

ODE

PDE

Applications

Other boundary conditions

See also

References