📂Partial Differential Equations

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

• Dirichlet boundary conditions

  $$u = 0 \quad \text{on } \partial \Omega$$

• Neumann boundary conditions

  $$\dfrac{\partial u}{\partial \nu} = 0 \quad \text{on } \partial \Omega$$

  Here, $\nu$ is the outward unit normal vector.

• Mixed Dirichlet-Neumann boundary conditions[^1]

  When $\partial \Omega$ contains two different closed sets $\Gamma_{1}$, $\Gamma_{2}$,

  $$\begin{align*} u = 0& \quad \text{on } \partial \Gamma_{1} \\ \dfrac{\partial u}{\partial \nu} = 0& \quad \text{on } \partial \Gamma_{2} \end{align*}$$

• Robin boundary conditions[^1]

  $$u + \dfrac{\partial u}{\partial \nu} = 0 \quad \text{on } \partial \Omega$$

  See Also

  • Initial Value Problem (IVP)

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