Dirichlet boundary condition

Differential equations
Scope
Fields Natural sciences Engineering Astronomy Physics Chemistry Biology Geology Applied mathematics Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics List of named differential equations
Classification
Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear By variable type Dependent and independent variables Autonomous Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation
Relation to processes Difference (discrete analogue) Stochastic Stochastic partial Delay
Solution
Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem
General topics Initial conditions Boundary values Dirichlet Neumann Robin Cauchy problem Wronskian Phase portrait Lyapunov / Asymptotic / Exponential stability Rate of convergence Series / Integral solutions Numerical integration Dirac delta function
Solution methods Inspection Method of characteristics Euler Exponential response formula Finite difference (Crank–Nicolson) Finite element Infinite element Finite volume Galerkin Petrov–Galerkin Green's function Integrating factor Integral transforms Perturbation theory Runge–Kutta Separation of variables Undetermined coefficients Variation of parameters
People
List Isaac Newton Gottfried Leibniz Jacob Bernoulli Leonhard Euler Józef Maria Hoene-Wroński Joseph Fourier Augustin-Louis Cauchy George Green Carl David Tolmé Runge Martin Kutta Rudolf Lipschitz Ernst Lindelöf Émile Picard Phyllis Nicolson John Crank
v t e

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 is 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.

Examples

ODE

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

For a partial differential equation, for example,

where ${\displaystyle \nabla ^{2}}$ 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

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

• In mechanical engineering and civil engineering (beam theory), where one end of a beam is held at a fixed position in space.
• In heat transfer, where a surface is held at a fixed temperature.
• In electrostatics, where a node of a circuit is held at a fixed voltage.
• In fluid dynamics, the no-slip condition 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.

See also

• Neumann boundary condition
• Robin boundary condition
• Boundary conditions in fluid dynamics

References