In mathematics, a Dirichlet boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values a solution is to take on the boundary of the domain. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ...

In the case of an ordinary differential equation such as

on the interval [0,1] the Dirichlet boundary conditions take the form

y(0) = α₁

y(1) = α₂

where α₁ and α₂ are given numbers.

For a partial differential equation on a domain

such as

Δy + y = 0

(Δ denotes the Laplacian), the Dirichlet boundary condition takes the form In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications. ...

where f is a known function defined on the boundary ∂Ω.

Dirichlet boundary conditions are perhaps the easiest to understand but there are many other conditions possible. For example, there is the Neumann boundary condition or the mixed boundary condition which is a combination of the Dirichlet and Neumann conditions. In mathematics, a Neumann boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values the derivative of a solution is to take on the boundary of the domain. ... In mathematics, a mixed boundary condition imposed on an ordinary differential equation or a partial differential equation give information about both the values of a function and the values of its derivative on the boundary of the domain. ...