Lazy composed PDE Operators

[ChrisRackauckas]

A nice feature would be able to compose these lazy operators. This will be difficult though. It would be killer to be able to support intermediates of scalars, functions, and AbstractArrays so that way something like

u_t = a(t,x)u_xx + b(t,x)u_x

will compose to a single operator on the LHS, or

u_t = (D*u_x)_x

where D is some vector of diffusion constants.

The composition should include the ability to combine with I, so that way I - gamma*L works.

Of course, we'd have to start with a simpler version, but this kind of composition would be absolutely awesome for using stiff integrators.

[shivin9]

@ChrisRackauckas Do you mean that the elements of the Transformation map will become functions now (say 2+x+sin(x))?

[ChrisRackauckas]

They probably shouldn't output the function, but instead should evaluate all of the proper functions so that way it outputs the solution.

If you have a design go ahead an implement, though I'm not entirely sure what the correct design for this is right now anyways. Instead of hanging on this, you might want to work on something more clear like boundary conditions while thinking about how to design this. It likely needs to be its own wrapper type which can wraper scalars, AbstractMatrixs, Functions, and LinearOperators, and then somehow "recall" how they should be put together.