remove t and D definitions

Author: ChrisRackauckas

docs/src/basics/Composition.md

@@ -21,7 +21,6 @@ using ModelingToolkit: t_nounits as t, D_nounits as D
 function decay(; name)
     @parameters a
     @variables x(t) f(t)
-    D = Differential(t)
     ODESystem([
             D(x) ~ -a * x + f
         ], t;
@@ -31,7 +30,6 @@ end
 @named decay1 = decay()
 @named decay2 = decay()
 
-D = Differential(t)
 connected = compose(
     ODESystem([decay2.f ~ decay1.x
                D(decay1.f) ~ 0], t; name = :connected), decay1, decay2)

docs/src/basics/Events.md

@@ -65,9 +65,10 @@ friction
 
 ```@example events
 using ModelingToolkit, OrdinaryDiffEq, Plots
+using ModelingToolkit: t_nounits as t, D_nounits as D
+
 function UnitMassWithFriction(k; name)
-    @variables t x(t)=0 v(t)=0
-    D = Differential(t)
+    @variables x(t)=0 v(t)=0
     eqs = [D(x) ~ v
            D(v) ~ sin(t) - k * sign(v)]
     ODESystem(eqs, t; continuous_events = [v ~ 0], name) # when v = 0 there is a discontinuity
@@ -87,8 +88,7 @@ an `affect!` on the state. We can model the same system using ModelingToolkit
 like this
 
 ```@example events
-@variables t x(t)=1 v(t)=0
-D = Differential(t)
+@variables x(t)=1 v(t)=0
 
 root_eqs = [x ~ 0]  # the event happens at the ground x(t) = 0
 affect = [v ~ -v] # the effect is that the velocity changes sign
@@ -108,8 +108,7 @@ plot(sol)
 Multiple events? No problem! This example models a bouncing ball in 2D that is enclosed by two walls at $y = \pm 1.5$.
 
 ```@example events
-@variables t x(t)=1 y(t)=0 vx(t)=0 vy(t)=2
-D = Differential(t)
+@variables x(t)=1 y(t)=0 vx(t)=0 vy(t)=2
 
 continuous_events = [[x ~ 0] => [vx ~ -vx]
                      [y ~ -1.5, y ~ 1.5] => [vy ~ -vy]]
@@ -229,7 +228,7 @@ Suppose we have a population of `N(t)` cells that can grow and die, and at time
 
 ```@example events
 @parameters M tinject α
-@variables t N(t)
+@variables N(t)
 Dₜ = Differential(t)
 eqs = [Dₜ(N) ~ α - N]
 

docs/src/basics/FAQ.md

@@ -102,9 +102,10 @@ end
 This error can come up after running `structural_simplify` on a system that generates dummy derivatives (i.e. variables with `ˍt`).  For example, here even though all the variables are defined with initial values, the `ODEProblem` generation will throw an error that defaults are missing from the variable map.
 
 ```
-@variables t
+using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
+
 sts = @variables x1(t)=0.0 x2(t)=0.0 x3(t)=0.0 x4(t)=0.0
-D = Differential(t)
 eqs = [x1 + x2 + 1 ~ 0
        x1 + x2 + x3 + 2 ~ 0
        x1 + D(x3) + x4 + 3 ~ 0
@@ -137,9 +138,9 @@ container type. For example:
 
 ```
 using ModelingToolkit, StaticArrays
-@variables t
+using ModelingToolkit: t_nounits as t, D_nounits as D
+
 sts = @variables x1(t)=0.0
-D = Differential(t)
 eqs = [D(x1) ~ 1.1 * x1]
 @mtkbuild sys = ODESystem(eqs, t)
 prob = ODEProblem{false}(sys, [], (0,1); u0_constructor = x->SVector(x...))

docs/src/basics/Linearization.md

@@ -21,9 +21,9 @@ The `linearize` function expects the user to specify the inputs ``u`` and the ou
 
 ```@example LINEARIZE
 using ModelingToolkit
-@variables t x(t)=0 y(t)=0 u(t)=0 r(t)=0
+using ModelingToolkit: t_nounits as t, D_nounits as D
+@variables x(t)=0 y(t)=0 u(t)=0 r(t)=0
 @parameters kp = 1
-D = Differential(t)
 
 eqs = [u ~ kp * (r - y) # P controller
        D(x) ~ -x + u    # First-order plant

docs/src/basics/Variable_metadata.md

@@ -50,8 +50,9 @@ current in a resistor. These variables sum up to zero in connections.
 
 ```@example connect
 using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
-@variables t, i(t) [connect = Flow]
+@variables i(t) [connect = Flow]
 @variables k(t) [connect = Stream]
 ```
 

docs/src/examples/higher_order.md

@@ -13,10 +13,10 @@ We utilize the derivative operator twice here to define the second order:
 
 ```@example orderlowering
 using ModelingToolkit, OrdinaryDiffEq
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @parameters σ ρ β
-@variables t x(t) y(t) z(t)
-D = Differential(t)
+@variables x(t) y(t) z(t)
 
 eqs = [D(D(x)) ~ σ * (y - x),
     D(y) ~ x * (ρ - z) - y,

docs/src/examples/perturbation.md

@@ -45,8 +45,9 @@ As the first ODE example, we have chosen a simple and well-behaved problem, whic
 with the initial conditions $x(0) = 0$, and $\dot{x}(0) = 1$. Note that for $\epsilon = 0$, this equation transforms back to the standard one. Let's start with defining the variables
 
 ```julia
+using ModelingToolkit: t_nounits as t, D_nounits as D
 n = 3
-@variables ϵ t y[1:n](t) ∂∂y[1:n](t)
+@variables ϵ y[1:n](t) ∂∂y[1:n](t)
 ```
 
 Next, we define $x$.
@@ -82,7 +83,6 @@ vals = solve_coef(eqs, ∂∂y)
 Our system of ODEs is forming. Now is the time to convert `∂∂`s to the correct **Symbolics.jl** form by substitution:
 
 ```julia
-D = Differential(t)
 subs = Dict(∂∂y[i] => D(D(y[i])) for i in eachindex(y))
 eqs = [substitute(first(v), subs) ~ substitute(last(v), subs) for v in vals]
 ```
@@ -147,7 +147,6 @@ vals = solve_coef(eqs, ∂∂y)
 Next, we need to replace `∂`s and `∂∂`s with their **Symbolics.jl** counterparts:
 
 ```julia
-D = Differential(t)
 subs1 = Dict(∂y[i] => D(y[i]) for i in eachindex(y))
 subs2 = Dict(∂∂y[i] => D(D(y[i])) for i in eachindex(y))
 subs = subs1 ∪ subs2

docs/src/examples/spring_mass.md

@@ -6,11 +6,9 @@ In this tutorial, we will build a simple component-based model of a spring-mass
 
 ```@example component
 using ModelingToolkit, Plots, DifferentialEquations, LinearAlgebra
+using ModelingToolkit: t_nounits as t, D_nounits as D
 using Symbolics: scalarize
 
-@variables t
-D = Differential(t)
-
 function Mass(; name, m = 1.0, xy = [0.0, 0.0], u = [0.0, 0.0])
     ps = @parameters m = m
     sts = @variables pos(t)[1:2]=xy v(t)[1:2]=u

docs/src/examples/tearing_parallelism.md

@@ -11,10 +11,9 @@ electrical circuits:
 
 ```@example tearing
 using ModelingToolkit, OrdinaryDiffEq
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 # Basic electric components
-@variables t
-const D = Differential(t)
 @connector function Pin(; name)
     @variables v(t)=1.0 i(t)=1.0 [connect = Flow]
     ODESystem(Equation[], t, [v, i], [], name = name)

docs/src/tutorials/acausal_components.md

@@ -21,8 +21,8 @@ equalities before solving. Let's see this in action.
 
 ```@example acausal
 using ModelingToolkit, Plots, DifferentialEquations
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
-@variables t
 @connector Pin begin
     v(t)
     i(t), [connect = Flow]
@@ -63,8 +63,6 @@ end
     end
 end
 
-D = Differential(t)
-
 @mtkmodel Capacitor begin
     @extend OnePort()
     @parameters begin
@@ -213,8 +211,6 @@ equations and unknowns and extend them with a new equation. Note that `v`, `i` a
 Using our knowledge of circuits, we similarly construct the `Capacitor`:
 
 ```@example acausal
-D = Differential(t)
-
 @mtkmodel Capacitor begin
     @extend OnePort()
     @parameters begin

docs/src/tutorials/bifurcation_diagram_computation.md

@@ -8,7 +8,9 @@ Let us first consider a simple `NonlinearSystem`:
 
 ```@example Bif1
 using ModelingToolkit
-@variables t x(t) y(t)
+using ModelingToolkit: t_nounits as t, D_nounits as D
+
+@variables x(t) y(t)
 @parameters μ α
 eqs = [0 ~ μ * x - x^3 + α * y,
     0 ~ -y]
@@ -87,10 +89,10 @@ It is also possible to use `ODESystem`s (rather than `NonlinearSystem`s) as inpu
 
 ```@example Bif2
 using BifurcationKit, ModelingToolkit, Plots
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
-@variables t x(t) y(t)
+@variables  x(t) y(t)
 @parameters μ
-D = Differential(t)
 eqs = [D(x) ~ μ * x - y - x * (x^2 + y^2),
     D(y) ~ x + μ * y - y * (x^2 + y^2)]
 @mtkbuild osys = ODESystem(eqs, t)

docs/src/tutorials/domain_connections.md

@@ -6,9 +6,7 @@ A domain in ModelingToolkit.jl is a network of connected components that share p
 
 ```@example domain
 using ModelingToolkit
-
-@parameters t
-D = Differential(t)
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @connector function HydraulicPort(; p_int, name)
     pars = @parameters begin

docs/src/tutorials/ode_modeling.md

@@ -20,9 +20,7 @@ But if you want to just see some code and run, here's an example:
 
 ```@example first-mtkmodel
 using ModelingToolkit
-
-@variables t
-D = Differential(t)
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @mtkmodel FOL begin
     @parameters begin
@@ -65,9 +63,7 @@ independent variable ``t`` is automatically added by ``@mtkmodel``.
 
 ```@example ode2
 using ModelingToolkit
-
-@variables t
-D = Differential(t)
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @mtkmodel FOL begin
     @parameters begin
@@ -109,6 +105,7 @@ intermediate variable `RHS`:
 
 ```@example ode2
 using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @mtkmodel FOL begin
     @parameters begin
@@ -118,9 +115,6 @@ using ModelingToolkit
         x(t) # dependent variables
         RHS(t)
     end
-    begin
-        D = Differential(t)
-    end
     @equations begin
         RHS ~ (1 - x) / τ
         D(x) ~ RHS
@@ -197,9 +191,6 @@ Obviously, one could use an explicit, symbolic function of time:
         x(t) # dependent variables
         f(t)
     end
-    begin
-        D = Differential(t)
-    end
     @equations begin
         f ~ sin(t)
         D(x) ~ (f - x) / τ
@@ -234,9 +225,6 @@ f_fun(t) = t >= 10 ? value_vector[end] : value_vector[Int(floor(t)) + 1]
     @structural_parameters begin
         h = 1
     end
-    begin
-        D = Differential(t)
-    end
     @equations begin
         f ~ f_fun(t)
         D(x) ~ (f - x) / τ
@@ -262,7 +250,7 @@ complex systems from simple ones is even more fun. Best practice for such a
 ```@example ode2
 function fol_factory(separate = false; name)
     @parameters τ
-    @variables t x(t) f(t) RHS(t)
+    @variables x(t) f(t) RHS(t)
 
     eqs = separate ? [RHS ~ (f - x) / τ,
         D(x) ~ RHS] :
@@ -358,10 +346,11 @@ In non-DSL definitions, one can pass `defaults` dictionary to set the initial gu
 
 ```@example ode3
 using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 function UnitstepFOLFactory(; name)
     @parameters τ
-    @variables t x(t)
+    @variables x(t)
     ODESystem(D(x) ~ (1 - x) / τ; name, defaults = Dict(x => 0.0, τ => 1.0))
 end
 ```

docs/src/tutorials/parameter_identifiability.md

@@ -30,9 +30,7 @@ We first define the parameters, variables, differential equations and the output
 
 ```@example SI
 using StructuralIdentifiability, ModelingToolkit
-
-@variables t
-D = Differential(t)
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @mtkmodel Biohydrogenation begin
     @variables begin
@@ -107,9 +105,7 @@ Global identifiability needs information about local identifiability first, but
 
 ```@example SI2
 using StructuralIdentifiability, ModelingToolkit
-
-@variables t
-D = Differential(t)
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @mtkmodel GoodwinOsc begin
     @parameters begin
@@ -150,9 +146,7 @@ Let us consider the same system but with two inputs, and we will find out identi
 
 ```@example SI3
 using StructuralIdentifiability, ModelingToolkit
-
-@variables t
-D = Differential(t)
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 @mtkmodel GoodwinOscillator begin
     @parameters begin

docs/src/tutorials/programmatically_generating.md

@@ -18,8 +18,9 @@ as demonstrated in the Symbolics.jl documentation. This looks like:
 
 ```@example scripting
 using Symbolics
-@variables t x(t) y(t) # Define variables
-D = Differential(t) # Define a differential operator
+using ModelingToolkit: t_nounits as t, D_nounits as D
+
+@variables x(t) y(t) # Define variables
 eqs = [D(x) ~ y
        D(y) ~ x] # Define an array of equations
 ```
@@ -32,11 +33,11 @@ defining PDESystem etc.
 
 ```@example scripting
 using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
-@variables t x(t)   # independent and dependent variables
+@variables x(t)   # independent and dependent variables
 @parameters τ       # parameters
 @constants h = 1    # constants
-D = Differential(t) # define an operator for the differentiation w.r.t. time
 eqs = [D(x) ~ (h - x) / τ] # create an array of equations
 
 # your first ODE, consisting of a single equation, indicated by ~

docs/src/tutorials/stochastic_diffeq.md

@@ -9,11 +9,11 @@ it to have multiplicative noise.
 
 ```@example SDE
 using ModelingToolkit, StochasticDiffEq
+using ModelingToolkit: t_nounits as t, D_nounits as D
 
 # Define some variables
 @parameters σ ρ β
-@variables t x(t) y(t) z(t)
-D = Differential(t)
+@variables x(t) y(t) z(t)
 
 eqs = [D(x) ~ σ * (y - x),
     D(y) ~ x * (ρ - z) - y,