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) who studied under Cauchy and succeeded Gauss at University of Göttingen.^[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.

In the case of an ordinary differential equation such as:

$\frac{d^2y}{dx^2} + 3 y = 1$

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

$\begin{align} y(0) &= \alpha _1 \\ y(1) &= \alpha _2 \end{align}$

where α₁ and α₂ are given numbers.

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

$\nabla^{2} y + y = 0\,$

where $\nabla^{2}$ denotes the Laplacian, the Dirichlet boundary condition takes 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.

See also

• Neumann boundary condition
• Mixed boundary condition
• Robin boundary condition
• Cauchy boundary condition

References

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