pull examples and complete homework 1

3 files changed, 103 insertions, 0 deletions

diff --git a/examples/FallingBall.jl b/examples/FallingBall.jl
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+#!/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)
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diff --git a/examples/FallingBall2.jl b/examples/FallingBall2.jl
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+#!/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)
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diff --git a/examples/FallingBall3.jl b/examples/FallingBall3.jl
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+++ b/examples/FallingBall3.jl
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+#!/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