6 files changed, 211 insertions, 0 deletions
diff --git a/examples/FallingBall.jl b/examples/FallingBall.jl
new file mode 100644
index 0000000..35d4b31
--- /dev/null
+++ b/examples/FallingBall.jl
@@ -0,0 +1,25 @@
+#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
+
+dt = 0.01 # time step in seconds
+g = 9.8 # acceleration of gravity in m/s^2
+
+function dynamics(y::Float64, v::Float64, t::Float64) 
+   for i in 1:100
+      y = y + v * dt
+      v = v - g * dt
+      t = t + dt
+   end
+
+   return y, v, t
+end
+
+y0 = 10.0 # initial position in meters
+v0 = 0.0 # initial velocity in m/s
+
+y, v, t = dynamics(y0, v0, 0.0) # evolave for 100 time steps
+
+println("\n\t Results")
+println("final time  = ", t)
+println("y = ", y, " and v = ", v)
+println("exact v = ", v0 - g * t)
+println("exact y = ", y0 + v0 * t - 0.5 * g * t^2.0)
+\ No newline at end of file
diff --git a/examples/FallingBall2.jl b/examples/FallingBall2.jl
new file mode 100644
index 0000000..535af6d
--- /dev/null
+++ b/examples/FallingBall2.jl
@@ -0,0 +1,44 @@
+#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
+
+using Plots # for plotting trajectory
+
+g = 9.8 # acceleration of gravity in m/s^2
+
+dt = 0.01 # time step in seconds
+t_final = 1.0 # final time of trajectory
+
+steps = Int64(t_final/dt) # number of time steps
+
+y = zeros(steps+1) # initial array of heights in meters
+v = zeros(steps+1) # initial array of velocities in m/s
+
+function dynamics!(y, v, t::Float64) # ! notation tells us that arguments will be modified
+   for i in 1:steps
+      y[i+1] = y[i] + v[i] * dt 
+      v[i+1] = v[i] - g * dt
+      #y[i+1] = y[i] + 0.5 * (v[i] + v[i+1]) * dt
+      t = t + dt
+   end
+
+   return t
+end
+
+y0 = 10.0 # initial position in meters
+v0 = 0.0 # initial velocity in m/s
+
+y[1] = y0
+v[1] = v0
+t = dynamics!(y, v, 0.0) # evolve forward in time
+
+println("\n\t Results")
+println("final time  = ", t)
+println("y = ", y[steps+1], " and v = ", v[steps+1])
+println("exact v = ", v0 - g * t)
+println("exact y = ", y0 + v0 * t - 0.5 * g * t^2.0)
+
+plot(y) # plot position as a function of time
+
+# energy = g * y + 0.5 * v .* v # here the mass = 1
+# println("initial energy = ", energy[1])
+# println("final energy = ", energy[steps+1])
+# plot(energy)
+\ No newline at end of file
diff --git a/examples/FallingBall3.jl b/examples/FallingBall3.jl
new file mode 100644
index 0000000..123a19a
--- /dev/null
+++ b/examples/FallingBall3.jl
@@ -0,0 +1,34 @@
+#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia
+
+using Plots # for plotting trajectory
+using DifferentialEquations # for solving ODEs
+
+g = 9.8 # acceleration of gravity in m/s^2
+
+t_final = 1.0 # final time of trajectory
+p = 0.0 # parameters (not used here)
+
+function tendency!(dyv::Vector{Float64}, yv::Vector{Float64}, p, t::Float64) # ! notation tells us that arguments will be modified
+   y = yv[1] # 2D phase space; use vcat(x, v) to combine 2 vectors
+   v = yv[2] # dy/dt = v
+   a = -g # dv/dt = -g
+
+   dyv[1] = v 
+   dyv[2] = a
+end
+
+y0 = 10.0 # initial position in meters
+v0 = 0.0 # initial velocity in m/s
+yv0 = [y0, v0] # initial condition in phase space 
+tspan = (0.0, t_final) # span of time to simulate
+
+prob = ODEProblem(tendency!, yv0, tspan, p) # specify ODE
+sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8) # solve using Tsit5 algorithm to specified accuracy
+
+println("\n\t Results")
+println("final time  = ", sol.t[end])
+println("y = ", sol[1, end], " and v = ", sol[2, end])
+println("exact v = ", v0 - g * t_final)
+println("exact y = ", y0 + v0 * t_final - 0.5 * g * t_final^2.0)
+
+plot(sol, idxs = (1)) # plot position as a function of time
diff --git a/hw1/1-3.jl b/hw1/1-3.jl
new file mode 100644
index 0000000..ad3f1b0
--- /dev/null
+++ b/hw1/1-3.jl
@@ -0,0 +1,40 @@
+using Plots # for plotting trajectory
+using DifferentialEquations # for solving ODEs
+
+
+# INITIAL CONDITIONALS AND PARAMETERS
+a = 10.0 # 1st drag coefficient
+b = 1.0 # 2nd drag coefficient (on v)
+v0 = 0.0 # initial velocity in m/s
+t_final = 10.0 # time of trajectory in s
+
+
+# EULER'S METHOD
+dt = 0.01 # time step
+steps = Int64(t_final/dt) # number of time steps
+
+v = zeros(steps+1) # initial array of velocities
+t = zeros(steps+1) # initial array of time intervals
+
+function dynamics!(v::Vector{Float64}, t::Vector{Float64}) 
+    for i in 1:steps
+        # equation: dv = dt(a - b*v)
+        dv = dt*(a - b*v[i])
+
+        v[i+1] = v[i] + dv
+        t[i+1] = t[i] + dt
+    end
+end
+
+# do the calculation, store into arrays
+v[1] = v0
+t[1] = 0.0
+dynamics!(v, t)
+
+# print the parameters & terminal velocity
+println("Parameters (a, b, v0, t_final): ", a, ", ", b, ", ", v0, ", ", t_final)
+println("Terminal velocity: ", v[end])
+
+# plot the velocities over time
+plot(t, v, xlabel="time (s)", ylabel="velocity (m/s)", title="Velocity vs. time (affected by drag, w/ a,b=10,3)", label="", lw=2, color=:blue)
+
diff --git a/hw1/1-6.jl b/hw1/1-6.jl
new file mode 100644
index 0000000..4704add
--- /dev/null
+++ b/hw1/1-6.jl
@@ -0,0 +1,68 @@
+using Plots # for plotting trajectory
+using DifferentialEquations # for solving ODEs
+
+
+# INITIAL CONDITIONALS AND PARAMETERS
+a = 10.0 # birth of new members
+b = 3.0 # death of members
+n0 = 10.0 # initial number of members
+t_final = 1.0 # final time of simulation
+p = 0.0 # parameters (not used here)
+dt = 0.01 # time step for euler's
+
+
+# EULER'S METHOD -> APPROXIMATE ANSWER
+steps = Int64(t_final/dt) # number of time steps
+
+n = zeros(steps+1) # initial array of members
+v = zeros(steps+1) # initial array of rate of change of members
+t = zeros(steps+1) # initial array of time intervals
+
+function dynamics!(n::Vector{Float64}, v::Vector{Float64}, t::Vector{Float64}) 
+    for i in 1:steps
+        # equation: dn = dt(aN - bN^2)
+        dn = dt*(a*n[i] - b*n[i]*n[i])
+        vn = dt*(a-2.0*b*n[i])
+
+        n[i+1] = n[i] + dn
+        v[i+1] = v[i] + vn
+        t[i+1] = t[i] + dt
+    end
+end
+
+# calcuate with current dt, store into arrays
+n[1] = n0
+v[1] = 0.0
+t[1] = 0.0
+dynamics!(n, v, t)
+
+
+# USING ODE SOLVER -> EXACT ANSWER
+
+function tendency!(dnv::Vector{Float64}, nv::Vector{Float64}, p, t::Float64) # ! notation tells us that arguments will be modified
+   n = nv[1] # 2D phase space; use vcat(x, v) to combine 2 vectors
+   v = nv[2] # dn/dt = v
+
+   dnv[1] = a*n - b*n*n
+   dnv[2] = a - 2.0*b*n
+end
+
+i0 = [n0, v0] # set initial conditions 
+tspan = (0.0, t_final) # span of time to simulate
+prob = ODEProblem(tendency!, i0, tspan, p) # specify ODE
+sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8) # solve using Tsit5 algorithm to specified accuracy
+n_exact = sol[1, :] # extract the population values over time
+
+
+# PLOTTING AND COMPARISON
+println("Parameters (a, b, n0, t_final): ", a, ", ", b, ", ", n0, ", ", t_final)
+println("Final population (at ", t_final, ") via Euler's Method:\t", n[end])
+println("Final population (at ", t_final, ") via ODE Solver:\t", n_exact[end])
+
+plot_title = "Population v. time (w/ n0,a,b= " * string(n0) * ", " * string(a) * ", " * string(b) * ")"
+plot(t, n, label="Euler's Method (dt = .01)", title=plot_title, lw=2, xlabel="time", ylabel="population")
+plot!(sol.t, n_exact, label="Exact Solution (ode solver)", lw=2) # plot!() to add to existing plot
+
+# NOTE: uncomment the two lines below if you also want to plot the next derativate
+# plot!(t, v, label="d^2n/dt^2 (Euler's Method)", lw=2)
+# plot!(sol.t, sol[2, :], label="d^2n/dt^2(ode solver)", lw=2)
+\ No newline at end of file
diff --git a/hw1/hw1-writeup.pdf b/hw1/hw1-writeup.pdf
new file mode 100644
index 0000000..cc8e38f
--- /dev/null
+++ b/hw1/hw1-writeup.pdf