In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
study of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, the Dirichlet (or first-type) boundary condition is a type of
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
, named after
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
(1805–1859). When imposed on an
ordinary or a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
, it specifies the values that a solution needs to take along the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
of the domain.
In
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
(FEM) analysis, ''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.
The question of finding solutions to such equations is known as the
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
. In applied sciences, a Dirichlet boundary condition may also be referred to as a fixed boundary condition.
Examples
ODE
For an
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
, for instance,
the Dirichlet boundary conditions on the interval take the form
where and are given numbers.
PDE
For a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
, for example,
where 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 function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, the Dirichlet boundary conditions on a domain take the form
where is a known
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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 that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
and
civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewage ...
(
beam theory
Beam may refer to:
Streams of particles or energy
*Light beam, or beam of light, a directional projection of light energy
**Laser beam
*Particle beam, a stream of charged or neutral particles
**Charged particle beam, a spatially localized grou ...
), where one end of a beam is held at a fixed position in space.
* In
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
, where a surface is held at a fixed temperature.
* In
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest (static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
, where a node of a circuit is held at a fixed voltage.
* In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, the
no-slip condition
In fluid dynamics, the no-slip condition for viscous fluids assumes that at a solid boundary, the fluid will have zero velocity relative to the boundary.
The fluid velocity at all fluid–solid boundaries is equal to that of the solid boundary. C ...
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 In mathematics, a Cauchy () boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Ca ...
and the
mixed boundary condition
In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary ...
. The latter is a combination of the Dirichlet and
Neumann conditions.
See also
*
Neumann boundary condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appl ...
*
Robin boundary condition
In mathematics, the Robin boundary condition (; properly ), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equatio ...
*
Boundary conditions in fluid dynamics
Boundary conditions in fluid dynamics are the set of constraints to boundary value problems in computational fluid dynamics. These boundary conditions include inlet boundary conditions, outlet boundary conditions, wall boundary conditions, constant ...
References
{{DEFAULTSORT:Dirichlet Boundary Condition
Boundary conditions