Dirichlet boundary condition

In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet (1805–1859).^[1] When imposed on an ordinary or a partial differential equation, it specifies the values a solution needs to take on the boundary of the domain. The question of finding solutions to such equations is known as the Dirichlet problem.

$y'' %2B y = 0~$

the Dirichlet boundary conditions on the interval $[a, \, b]$ take the form:

$y(a)= \alpha \ \text{and} \ y(b) = \beta$

where $\alpha$ and $\beta$ are given numbers.

$\nabla^2 y %2B y = 0$

where $\nabla^2$ denotes the Laplacian, the Dirichlet boundary conditions on a domain $\Omega \subset \mathbb{R}^n$ take the form:

$y(x) = f(x) \quad \forall x \in \partial\Omega$

where f is a known function defined on the boundary $\partial\Omega$ .

Many other boundary conditions are possible. For example, there is the Cauchy boundary condition, or the mixed boundary condition which is a combination of the Dirichlet and Neumann conditions.

References

^ Cheng, A. and D. T. Cheng (2005). Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements, 29, 268–302.