Ciro Santilli @cirosantilli 37

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.

Continuous version of the Fourier series.

Can be used to represent functions that are not periodic: math.stackexchange.com/questions/221137/what-is-the-difference-between-fourier-series-and-fourier-transformation while the Fourier series is only for periodic functions.

Of course, every function defined on a finite line segment (i.e. a compact space).

Therefore, the Fourier transform can be seen as a generalization of the Fourier series that can also decompose functions defined on the entire real line.

As a more concrete example, just like the Fourier series is how you solve the heat equation on a line segment with Dirichlet boundary conditions as shown at: Section "Solving partial differential equations with the Fourier series", the Fourier transform is what you need to solve the problem when the domain is the entire real line.

1-dimensional heat equation example with Dirichlet boundary condition

2-dimensional heat equation example with Dirichlet boundary condition:

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

Examples:

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:

www.cns.gatech.edu/~predrag/courses/PHYS-6124-12/StGoChap6.pdf 6.1 "Classification of PDE's" clarifies which boundary conditions are needed for existence and uniqueness of each type of second order of PDE: