refactor: format

Author: AayushSabharwal

docs/src/examples/perturbation.md

@@ -5,46 +5,57 @@ In the [Mixed Symbolic-Numeric Perturbation Theory tutorial](https://symbolics.j
 ## Free fall in a varying gravitational field
 
 Our first ODE example is a well-known physics problem: what is the altitude $x(t)$ of an object (say, a ball or a rocket) thrown vertically with initial velocity $ẋ(0)$ from the surface of a planet with mass $M$ and radius $R$? According to Newton's second law and law of gravity, it is the solution of the ODE
+
 ```math
 ẍ = -\frac{GM}{(R+x)^2} = -\frac{GM}{R^2} \frac{1}{\left(1+ϵ\frac{x}{R}\right)^2}.
 ```
+
 In the last equality, we introduced a perturbative expansion parameter $ϵ$. When $ϵ=1$, we recover the original problem. When $ϵ=0$, the problem reduces to the trivial problem $ẍ = -g$ with constant gravitational acceleration $g = GM/R^2$ and solution $x(t) = x(0) + ẋ(0) t - \frac{1}{2} g t^2$. This is a good setup for perturbation theory.
 
 To make the problem dimensionless, we redefine $x \leftarrow x / R$ and $t \leftarrow t / \sqrt{R^3/GM}$. Then the ODE becomes
+
 ```@example perturbation
 using ModelingToolkit
 using ModelingToolkit: t_nounits as t, D_nounits as D
 @variables ϵ x(t)
-eq = D(D(x)) ~ -(1 + ϵ*x)^(-2)
+eq = D(D(x)) ~ -(1 + ϵ * x)^(-2)
 ```
 
 Next, expand $x(t)$ in a series up to second order in $ϵ$:
+
 ```@example perturbation
 using Symbolics
 @variables y(t)[0:2] # coefficients
 x_series = series(y, ϵ)
 ```
 
 Insert this into the equation and collect perturbed equations to each order:
+
 ```@example perturbation
 eq_pert = substitute(eq, x => x_series)
 eqs_pert = taylor_coeff(eq_pert, ϵ, 0:2)
 ```
+
 !!! note
+    
     The 0-th order equation can be solved analytically, but ModelingToolkit does currently not feature automatic analytical solution of ODEs, so we proceed with solving it numerically.
-These are the ODEs we want to solve. Now construct an `ODESystem`, which automatically inserts dummy derivatives for the velocities:
+    These are the ODEs we want to solve. Now construct an `ODESystem`, which automatically inserts dummy derivatives for the velocities:
+
 ```@example perturbation
 @mtkbuild sys = ODESystem(eqs_pert, t)
 ```
 
 To solve the `ODESystem`, we generate an `ODEProblem` with initial conditions $x(0) = 0$, and $ẋ(0) = 1$, and solve it:
+
 ```@example perturbation
 using DifferentialEquations
 u0 = Dict([unknowns(sys) .=> 0.0; D(y[0]) => 1.0]) # nonzero initial velocity
 prob = ODEProblem(sys, u0, (0.0, 3.0))
 sol = solve(prob)
 ```
+
 This is the solution for the coefficients in the series for $x(t)$ and their derivatives. Finally, we calculate the solution to the original problem by summing the series for different $ϵ$:
+
 ```@example perturbation
 using Plots
 p = plot()
@@ -53,6 +64,7 @@ for ϵᵢ in 0.0:0.1:1.0
 end
 p
 ```
+
 This makes sense: for larger $ϵ$, gravity weakens with altitude, and the trajectory goes higher for a fixed initial velocity.
 
 An advantage of the perturbative method is that we run the ODE solver only once and calculate trajectories for several $ϵ$ for free. Had we solved the full unperturbed ODE directly, we would need to do repeat it for every $ϵ$.
@@ -62,19 +74,23 @@ An advantage of the perturbative method is that we run the ODE solver only once
 Our second example applies perturbation theory to nonlinear oscillators -- a very important class of problems. As we will see, perturbation theory has difficulty providing a good solution to this problem, but the process is nevertheless instructive. This example closely follows chapter 7.6 of *Nonlinear Dynamics and Chaos* by Steven Strogatz.
 
 The goal is to solve the ODE
+
 ```@example perturbation
-eq = D(D(x)) + 2*ϵ*D(x) + x ~ 0
+eq = D(D(x)) + 2 * ϵ * D(x) + x ~ 0
 ```
+
 with initial conditions $x(0) = 0$ and $ẋ(0) = 1$. With $ϵ = 0$, the problem reduces to the simple linear harmonic oscillator with the exact solution $x(t) = \sin(t)$.
 
 We follow the same steps as in the previous example to construct the `ODESystem`:
+
 ```@example perturbation
 eq_pert = substitute(eq, x => x_series)
 eqs_pert = taylor_coeff(eq_pert, ϵ, 0:2)
 @mtkbuild sys = ODESystem(eqs_pert, t)
 ```
 
 We solve and plot it as in the previous example, and compare the solution with $ϵ=0.1$ to the exact solution $x(t, ϵ) = e^{-ϵ t} \sin(\sqrt{(1-ϵ^2)}\,t) / \sqrt{1-ϵ^2}$ of the unperturbed equation:
+
 ```@example perturbation
 u0 = Dict([unknowns(sys) .=> 0.0; D(y[0]) => 1.0]) # nonzero initial velocity
 prob = ODEProblem(sys, u0, (0.0, 50.0))

src/systems/abstractsystem.jl

@@ -3001,8 +3001,8 @@ By default, the resulting system inherits `sys`'s name and description.
 See also [`compose`](@ref).
 """
 function extend(sys::AbstractSystem, basesys::AbstractSystem;
-    name::Symbol = nameof(sys), description = description(sys),
-    gui_metadata = get_gui_metadata(sys))
+        name::Symbol = nameof(sys), description = description(sys),
+        gui_metadata = get_gui_metadata(sys))
     T = SciMLBase.parameterless_type(basesys)
     ivs = independent_variables(basesys)
     if !(sys isa T)