Support discretization when lhs/rhs equal to zero

[akashkgarg]

When the equation is setup like so:

Dt(u(x,t)) + u(x,t)*Dx(u(x,t)) + α*Dx2(u(x,t)) + β*Dx3(u(x,t)) + γ*Dx4(u(x,t)) ~ 0

an InvalidSystemException is thrown:

ERROR: InvalidSystemException: The derivative variable must be isolated to the left-hand side of the equation like `Differential(t)(u₂(t)) ~ ...`.
 Got 0 ~ 137.50000000000009u₃(t) - (96.87499999999991u₂(t)) - (33.3333333333334u₄(t)) - (7.2916666666667105u₅(t)) - Differential(t)(u₂(t)) - (u₂(t)*(3.750000000000008u₃(t) + 0.20833333333333368u₅(t) + 0.2499999999999997(sech(5.0 + t)^2)*(3tanh(5.0 + t) - 1)*(7.5 + 7.5tanh(5.0 + t)) - (2.7083333333333344u₂(t)) - (1.250000000000007u₄(t)))) - (19.583333333333307(sech(5.0 + t)^2)*(3tanh(5.0 + t) - 1)*(7.5 + 7.5tanh(5.0 + t))).   

The current work around is re-writing the equation as:

Dt(u(x,t)) ~ -u(x,t)*Dx(u(x,t)) - α*Dx2(u(x,t)) - β*Dx3(u(x,t)) - γ*Dx4(u(x,t))

Here is the full script to reproduce:

    using ModelingToolkit,DiffEqOperators,LinearAlgebra,OrdinaryDiffEq

    @parameters x, t
    @variables u(..)
    Dt = Differential(t)
    Dx = Differential(x)
    Dx2 = Differential(x)^2
    Dx3 = Differential(x)^3
    Dx4 = Differential(x)^4

    α = 1
    β = 4
    γ = 1
    eq = Dt(u(x,t)) + u(x,t)*Dx(u(x,t)) + α*Dx2(u(x,t)) + β*Dx3(u(x,t)) + γ*Dx4(u(x,t)) ~ 0

    du(x,t;z = -x/2+t) = 15/2*(tanh(z) + 1)*(3*tanh(z) - 1)*sech(z)^2

    bcs = [u(x,0) ~ u_analytic(x,0),
           Dx(u(-10,t)) ~ du(-10,t),
           Dx(u(10,t)) ~ du(10,t)]

    # Space and time domains
    domains = [x ∈ IntervalDomain(-10.0,10.0),
               t ∈ IntervalDomain(0.0,0.5)]
    # Discretization
    dx = 0.4; dt = 0.2

    discretization = MOLFiniteDifference([x=>dx],t;centered_order=4)
    pdesys = PDESystem(eq,bcs,domains,[x,t],[u(x,t)])
    prob = discretize(pdesys,discretization) # <-- fails here. 

[ChrisRackauckas]

This is an MOLFiniteDifference issue, not an MTK issue. For example, NeuralPDE.jl is just fine with this. DiffEqOperators.jl is what needs to extend its parsing to be more general, so I'm transferring the issue there.

I think what it needs to do is check for the undiscretized indepedndent variable, move all derivatives of that over, validate the equation, throw an error if it cannot discretize it, and then continue.

[xtalax]

Somewhere along the way this got fixed