If you're getting some cold feet to jump in to DiffEq land, here are some handcrafted differential equations mini problems to hold your hand along the beginning of your journey.

\[ f(t,u) = \frac{du}{dt} \]

The Radioactive decay problem is the first order linear ODE problem of an exponential with a negative coefficient, which represents the half-life of the process in question. Should the coefficient be positive, this would represent a population growth equation.

using OrdinaryDiffEq, Plots gr() #Half-life of Carbon-14 is 5,730 years. C₁ = 5.730 #Setup u₀ = 1.0 tspan = (0.0, 1.0) #Define the problem radioactivedecay(u,p,t) = -C₁*u #Pass to solver prob = ODEProblem(radioactivedecay,u₀,tspan) sol = solve(prob,Tsit5()) #Plot plot(sol,linewidth=2,title ="Carbon-14 half-life", xaxis = "Time in thousands of years", yaxis = "Percentage left", label = "Numerical Solution") plot!(sol.t, t->exp(-C₁*t),lw=3,ls=:dash,label="Analytical Solution")

Another classical example is the harmonic oscillator, given by

\[ \ddot{x} + \omega^2 x = 0 \]

with the known analytical solution

\[ \begin{align*} x(t) &= A\cos(\omega t - \phi) \\ v(t) &= -A\omega\sin(\omega t - \phi), \end{align*} \]

where

\[ A = \sqrt{c_1 + c_2} \qquad\text{and}\qquad \tan \phi = \frac{c_2}{c_1} \]

with $c_1, c_2$ constants determined by the initial conditions such that $c_1$ is the initial position and $\omega c_2$ is the initial velocity.

Instead of transforming this to a system of ODEs to solve with `ODEProblem`

, we can use `SecondOrderODEProblem`

as follows.

# Simple Harmonic Oscillator Problem using OrdinaryDiffEq, Plots #Parameters ω = 1 #Initial Conditions x₀ = [0.0] dx₀ = [π/2] tspan = (0.0, 2π) ϕ = atan((dx₀[1]/ω)/x₀[1]) A = √(x₀[1]^2 + dx₀[1]^2) #Define the problem function harmonicoscillator(ddu,du,u,ω,t) ddu .= -ω^2 * u end #Pass to solvers prob = SecondOrderODEProblem(harmonicoscillator, dx₀, x₀, tspan, ω) sol = solve(prob, DPRKN6()) #Plot plot(sol, vars=[2,1], linewidth=2, title ="Simple Harmonic Oscillator", xaxis = "Time", yaxis = "Elongation", label = ["x" "dx"]) plot!(t->A*cos(ω*t-ϕ), lw=3, ls=:dash, label="Analytical Solution x") plot!(t->-A*ω*sin(ω*t-ϕ), lw=3, ls=:dash, label="Analytical Solution dx")

Note that the order of the variables (and initial conditions) is `dx`

, `x`

. Thus, if we want the first series to be `x`

, we have to flip the order with `vars=[2,1]`

.

We will start by solving the pendulum problem. In the physics class, we often solve this problem by small angle approximation, i.e. $ sin(\theta) \approx \theta$, because otherwise, we get an elliptic integral which doesn't have an analytic solution. The linearized form is

\[ \ddot{\theta} + \frac{g}{L}{\theta} = 0 \]

But we have numerical ODE solvers! Why not solve the *real* pendulum?

\[ \ddot{\theta} + \frac{g}{L}{\sin(\theta)} = 0 \]

Notice that now we have a second order ODE. In order to use the same method as above, we nee to transform it into a system of first order ODEs by employing the notation $d\theta = \dot{\theta}$.

\[ \begin{align*} &\dot{\theta} = d{\theta} \\ &\dot{d\theta} = - \frac{g}{L}{\sin(\theta)} \end{align*} \]

# Simple Pendulum Problem using OrdinaryDiffEq, Plots #Constants const g = 9.81 L = 1.0 #Initial Conditions u₀ = [0,π/2] tspan = (0.0,6.3) #Define the problem function simplependulum(du,u,p,t) θ = u[1] dθ = u[2] du[1] = dθ du[2] = -(g/L)*sin(θ) end #Pass to solvers prob = ODEProblem(simplependulum, u₀, tspan) sol = solve(prob,Tsit5()) #Plot plot(sol,linewidth=2,title ="Simple Pendulum Problem", xaxis = "Time", yaxis = "Height", label = ["\\theta" "d\\theta"])

So now we know that behaviour of the position versus time. However, it will be useful to us to look at the phase space of the pendulum, i.e., and representation of all possible states of the system in question (the pendulum) by looking at its velocity and position. Phase space analysis is ubiquitous in the analysis of dynamical systems, and thus we will provide a few facilities for it.

p = plot(sol,vars = (1,2), xlims = (-9,9), title = "Phase Space Plot", xaxis = "Velocity", yaxis = "Position", leg=false) function phase_plot(prob, u0, p, tspan=2pi) _prob = ODEProblem(prob.f,u0,(0.0,tspan)) sol = solve(_prob,Vern9()) # Use Vern9 solver for higher accuracy plot!(p,sol,vars = (1,2), xlims = nothing, ylims = nothing) end for i in -4pi:pi/2:4π for j in -4pi:pi/2:4π phase_plot(prob, [j,i], p) end end plot(p,xlims = (-9,9))