Neumann Boundary Conditions 📂Partial Differential Equations

Neumann Boundary Conditions

Definition¹

Let’s assume a partial differential equation is given, defined on an open set $\Omega$. The following boundary condition is called the Neumann boundary condition. The problem of finding the solution to the partial differential equation with the Neumann boundary condition is referred to as the Neumann problem.

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

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

Description

Nonhomogeneous Condition

The following boundary condition is sometimes referred to as the nonhomogeneous Neumann condition, but in most cases, it is not meticulously noted whether it is homogeneous or nonhomogeneous.

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

Example

For instance, solving the Neumann problem in the Poisson’s equation involves finding a $u$ that satisfies the following condition.

$$ \left\{ \begin{align*} -\Delta u = f & \quad \text{in } \Omega \\ \dfrac{\partial u}{\partial \nu} = 0 & \quad \text{on }\partial \Omega \end{align*} \right. $$