Dirichlet boundary condition
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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.
In the case of an ordinary differential equation such as
- <math>
\frac{d^2y}{dx^2} + 3 y = 1 <math>
on the interval <math>[0,1]<math> the Dirichlet boundary conditions take the form
- <math>y(0) = \alpha_1<math>
- <math>y(1) = \alpha_2<math>
where <math>\alpha_1<math> and <math>\alpha_2<math> are given numbers.
For a partial differential equation on a domain
- <math>\Omega\subset R^n<math>
such as
- <math>
\Delta y + y = 0 <math>
(<math>\Delta<math> denotes the Laplacian), the Dirichlet boundary condition takes the form
- <math>
y(x) = f(x) \quad \forall x \in \partial\Omega <math>
where <math>f<math> 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.