# Boundary condition

## Initial 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.

## Dirichlet boundary condition

Specifies fixed values.
Numerical examples:

## Neumann boundary condition

Specifies the derivative in a direction normal to the boundary.

## Cauchy boundary condition

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

## Robin boundary condition

Linear combination of a Dirichlet boundary condition and Neumann boundary condition at each point of the boundary.
Examples:
• heat equation when metal plaque is immersed in a large external environment of fixed temperature.
In this case, the normal derivative at the boundary is proportional to the difference between the temperature of the boundary and the fixed temperature of the external environment.
The result as time tends to infinity is that the temperature of the plaque tends to that of the environment.

## Open boundary condition

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:

## Mixed boundary condition

Multiple boundary conditions for different parts of the boundary.

## Time dependent boundary condition

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.