Operation vs Variable

[YingboMa]

I think it might be a good idea to use Operations everywhere. This should solve #415. The problem is that the states are variables but the equations can have an Operation that behaves like a "variable", i.e.

julia> sys = ODESystem(eqs, t, [u,Dk(u)], [k])
ODESystem(Equation[Equation(derivative(u(t), t), -k * u(t)), Equation(derivative(derivative(u(t), k), t), -(u(t)) - k * derivative(u(t), k))], t, Variable[u, uˍk], Variable[k], Base.RefValue{Array{Expression,1}}(Expression[]), Base.RefValue{Array{Expression,2}}(Array{Expression}(undef,0,0)), Base.RefValue{Array{Expression,2}}(Array{Expression}(undef,0,0)), Base.RefValue{Array{Expression,2}}(Array{Expression}(undef,0,0)), Symbol("##ODESystem#294"), ODESystem[])

julia> sys.eqs
2-element Array{Equation,1}:
 Equation(derivative(u(t), t), -k * u(t))
 Equation(derivative(derivative(u(t), k), t), -(u(t)) - k * derivative(u(t), k))

julia> sys.states
2-element Array{Variable,1}:
 u
 uˍk

Though this could break A LOT OF internal assumptions, and it will take some time to do this refactoring.

[YingboMa]

Using Operations everywhere could make the order lowering code a lot nicer, too.

[ChrisRackauckas]

Operations everywhere might make a lot of things nicer. I know @isaacsas was thinking about it. I think someone just has to do it for us to see whether it's better. I think it would probably be a change for the good.

[isaacsas]

This would be great, assuming it would not cause performance issues when checking equality or such. Something like #312 would be good to have too for Dictionary usage, which gives a big speed up for large systems over having to search array mappings. (Though testing it out right now I guess Operation to integer mappings in Dicts already work, which might be all I would need.)

[ChrisRackauckas]

In a similar vein, I wonder if we even need Constant(x), since it can always be represented as the Operation identity(x)

[ChrisRackauckas]

I don't think we can fix #312 all of the way without fixing this.

[shashi]

Closed by #518