Boundary Value Problems in Partial Differential Equations
Definition
Given a partial differential equation defined in an open set $\Omega$, let’s assume the values of the unknown $u$ are given on the boundary $\partial \Omega$ of $\Omega$. This is called a boundary condition. The partial differential equation together with the boundary condition is referred to as a boundary value problem.
Description
The abbreviation BVP is commonly used.
To solve a boundary value problem means to find a solution $u$ that satisfies the boundary conditions within the given partial differential equation.
Examples
$$ u = 0 \quad \text{on } \partial \Omega $$
$$ \dfrac{\partial u}{\partial \nu} = 0 \quad \text{on } \partial \Omega $$
Here, $\nu$ is the outward unit normal vector.
Mixed Dirichlet-Neumann boundary conditions[^1]
When $\partial \Omega$ contains two different closed sets $\Gamma_{1}$, $\Gamma_{2}$,
$$ \begin{align*} u = 0& \quad \text{on } \partial \Gamma_{1} \\ \dfrac{\partial u}{\partial \nu} = 0& \quad \text{on } \partial \Gamma_{2} \end{align*} $$
$$ u + \dfrac{\partial u}{\partial \nu} = 0 \quad \text{on } \partial \Omega $$
See Also
□re