Shell R code

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+.Rproj.user
+.Rhistory
+.RData
+.Ruserdata

CS-7863-Sci-Stat-Proj-4.Rproj

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+
+RestoreWorkspace: Default
+SaveWorkspace: Default
+AlwaysSaveHistory: Default
+
+EnableCodeIndexing: Yes
+UseSpacesForTab: Yes
+NumSpacesForTab: 2
+Encoding: UTF-8
+
+RnwWeave: Sweave
+LaTeX: pdfLaTeX

Schrick-Noah_Homework-4.R

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+# Project 4 for the University of Tulsa's CS-7863 Sci-Stat Course
+# Higher Order Differential Equations and Shooting Method
+# Professor: Dr. McKinney, Spring 2023
+# Noah L. Schrick - 1492657
+
+## 1. Transform the second order ODE into a system of two first order ODEs
+
+# use ode45 or rk4sys to solve for the motion of the plane pendulum 
+tmin <- 0
+tmax <- 7
+init_angle <- pi/4
+init_ang_spd <- 0
+g <- 9.81
+L <- 1
+
+# a. Plot
+plot(pend.sol[,1],pend.sol[,2],type="l",col="blue", 
+     ylim=c(-2.3,2.3), xlab="time",ylab="amplitude")
+par(new=T)
+lines(pend.sol[,1],pend.sol[,3],type="l",col="red")
+abline(h=0)
+legend("topleft", # coordinates, "topleft" etc
+       c("angle","omega"), # label
+       lty=c(1,1), # line 
+       lwd=c(2.5,2.5), # weight
+       #cex=.8, 
+       bty="n", # no box
+       col=c("blue","red") # color
+)
+
+# b. Overlay a plot of the small-angle approx
+
+## 2. Use ode45 to solve the damped harmonic oscillator 
+tmin <- 0
+tmax <- 10
+yo <- 1
+ybaro <- 0
+
+# a
+m <- 1
+c <- 1
+k <- 2
+
+# plot pos vs time
+
+# b
+m <- 1
+c <- 2
+k <- 1
+
+# plot pos vs time
+
+## 3. Use ode45/rk4sys to solve for the traj of a projectile thrown vertically 
+
+# a. IVP
+tmin <- 0
+tmax <- 2.04
+xo <- 0
+vo <- 10
+
+# b. BVP
+yo <- 0
+ymax <- 0
+
+library(pracma)
+projectile.f <- function(t,y){
+  g <- 9.81
+  v <- y[2]  # y1dot
+  a <- -g  # could be any f, y'' = f(t,y)
+  matrix(c(v,a))
+}
+
+proj.obj <- function(v0, y0=0, tfinal){
+  # minimize w.r.t. v0
+  proj.sol <- ode45(projectile.f, 
+                    y=c(y0, v0), t0=0, tfinal=tfinal)
+  final_index <- length(proj.sol$t)
+  yf <- proj.sol$y[final_index,1]  # want equal to right boundary 
+  log(abs(yf)) # minimize this
+}
+
+# user specifies tfinal and yfinal for BVP
+v_best <- optimize(proj.obj, 
+                   interval=c(1,100), #bisect-esque interval
+                   tol=1e-10, 
+                   y0=0, tfinal=10) # un-optimized obj params
+v_best$minimum  # best v0
+
+best.sol <- rk4sys(projectile.f, a=0, b=10, y0=c(0, v_best$minimum),
+                   n=20)  # 20 integration stepstmax 
+
+# c. Shooting method for damped oscillator with perturbation parameter
+yo <- 0
+y1 <- 1
+ymin <- 0
+ymax <- 2
+
+pfirst <- 0.5
+psec <- 0.05
+
+# analytical sol
+pthird <- 0
+
+## 4. Position of the earth and moon
+# a. Plotly
+G <- 6.673e-11   # m^3 kg-1 s^-2
+M_S <- 1.9891e30 # sun kg
+M_E <- 5.98e24   # earth kg
+M_m <- 7.32e22   # moon kg
+mu_sun <- G*M_S*(86400^2)/1e9 #  km^3/days^2
+
+# t0: jan 1, 1999 00:00:00am
+x0_earth <- c(-27115219762.4, 132888652547.0, 57651255508.0)/1e3 # km
+v0_earth <- c(-29794.2199907, -4924.33333333,-2135.59540741)*(86400)/1e3 
+# m/s -> km/day
+
+x0_moon <- c(-27083318944, 133232649728, 57770257344)/1e3 # km
+v0_moon <- c(-30864.2207031, -4835.03349304, -2042.89546204)*(86400)/1e3 
+# m/s -> km/day
+
+# b. Find eclipses
+
+# c. keep orbiter at L2 Lagrange point for a year, ignoring the moon's effect