Boundary condition

Basically a subset of the boundary condition for when one of the parameters is time and we are specifying values for the time 0.

Specifies fixed values.

Can be used for elliptic partial differential equations and parabolic partial differential equations.

Numerical examples:

Specifies the derivative in a direction normal to the boundary.

Can be used for elliptic partial differential equations and parabolic partial differential equations.

Sets both a Dirichlet boundary condition and a Neumann boundary condition for a single part of the boundary.

Can be used for hyperbolic partial differential equations.

We understand intuitively that this imposes stricter requirements on solutions, which makes it easier to guarantee uniqueness, but also harder to have existence. TODO intuitively why hyperbolic need this extra level of restriction.

Linear combination of a Dirichlet boundary condition and Neumann boundary condition at each point of the boundary.

Examples:

In the context of wave-like equations, an open-boundary condition is one that "lets the wave go through without reflection".

This condition is very useful when we want to simulate infinite domains with a numerical method. Ciro Santilli wants to do this all the time when trying to come up with demos for his physics writings.

Here are some resources that cover such boundary conditions:

Multiple boundary conditions for different parts of the boundary.

Most commonly, boundary conditions such as the Dirichlet boundary condition are taken to be fixed values in time.

But it also makes sense to think about cases where those values vary in time.

Some bibliography: