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In the case of an ordinary differential equation such as

:
rac{d^2y}{dx^2} + 3 y = 1


on the interval {Link without Title} the Dirichlet boundary conditions take the form

:y(0) = \alpha_1
:y(1) = \alpha_2

where \alpha_1 and \alpha_2 are given numbers.

For a partial differential equation on a domain
:\Omega\subset R^n
such as

:
\Delta y + y = 0


(\Delta denotes the Laplacian ), the Dirichlet boundary condition takes the form

:
y(x) = f(x) \quad orall x \in \partial\Omega


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.