Merge pull request #3255 from ArnoStrouwen/docs6

update higher order documentation to modern MTK

Author: ChrisRackauckas

docs/src/examples/higher_order.md

@@ -3,7 +3,7 @@
 ModelingToolkit has a system for transformations of mathematical
 systems. These transformations allow for symbolically changing
 the representation of the model to problems that are easier to
-numerically solve. One simple to demonstrate transformation is the
+numerically solve. One simple to demonstrate transformation, is
 `structural_simplify`, which does a lot of tricks, one being the
 transformation that turns an Nth order ODE into N
 coupled 1st order ODEs.
@@ -15,16 +15,28 @@ We utilize the derivative operator twice here to define the second order:
 using ModelingToolkit, OrdinaryDiffEq
 using ModelingToolkit: t_nounits as t, D_nounits as D
 
-@parameters σ ρ β
-@variables x(t) y(t) z(t)
-
-eqs = [D(D(x)) ~ σ * (y - x),
-    D(y) ~ x * (ρ - z) - y,
-    D(z) ~ x * y - β * z]
-
-@named sys = ODESystem(eqs, t)
+@mtkmodel SECOND_ORDER begin
+    @parameters begin
+        σ = 28.0
+        ρ = 10.0
+        β = 8 / 3
+    end
+    @variables begin
+        x(t) = 1.0
+        y(t) = 0.0
+        z(t) = 0.0
+    end
+    @equations begin
+        D(D(x)) ~ σ * (y - x)
+        D(y) ~ x * (ρ - z) - y
+        D(z) ~ x * y - β * z
+    end
+end
+@mtkbuild sys = SECOND_ORDER()
 ```
 
+The second order ODE has been automatically transformed to two first order ODEs.
+
 Note that we could've used an alternative syntax for 2nd order, i.e.
 `D = Differential(t)^2` and then `D(x)` would be the second derivative,
 and this syntax extends to `N`-th order. Also, we can use `*` or `∘` to compose
@@ -33,28 +45,17 @@ and this syntax extends to `N`-th order. Also, we can use `*` or `∘` to compos
 Now let's transform this into the `ODESystem` of first order components.
 We do this by calling `structural_simplify`:
 
-```@example orderlowering
-sys = structural_simplify(sys)
-```
-
 Now we can directly numerically solve the lowered system. Note that,
 following the original problem, the solution requires knowing the
-initial condition for `x'`, and thus we include that in our input
-specification:
+initial condition for both `x` and `D(x)`.
+The former already got assigned a default value in the `@mtkmodel`,
+but we still have to provide a value for the latter.
 
 ```@example orderlowering
-u0 = [D(x) => 2.0,
-    x => 1.0,
-    y => 0.0,
-    z => 0.0]
-
-p = [σ => 28.0,
-    ρ => 10.0,
-    β => 8 / 3]
-
+u0 = [D(sys.x) => 2.0]
 tspan = (0.0, 100.0)
-prob = ODEProblem(sys, u0, tspan, p, jac = true)
+prob = ODEProblem(sys, u0, tspan, [], jac = true)
 sol = solve(prob, Tsit5())
-using Plots;
-plot(sol, idxs = (x, y));
+using Plots
+plot(sol, idxs = (sys.x, sys.y))
 ```