Part 1a with forward approximation method

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CS-7863-Sci-Stat-Proj-3.Rproj

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Schrick-Noah_Homework-3.R

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+# Project 3 for the University of Tulsa's CS-7863 Sci-Stat Course
+# Numerical Ordinary Differential Equations
+# Professor: Dr. McKinney, Spring 2023
+# Noah L. Schrick - 1492657
+
+## 1. Approximate deriv. of sin(x) at x=pi from h=10^-1 -> 10^-20
+forward.approx <- function(func, x, h){
+  approx <- (func(x+h) - func(x))/h
+}
+
+forward.approx.table <- matrix(nrow = 0, ncol = 3)  
+colnames(forward.approx.table) <- c("h", "approximation", "error")
+for(h in 10^(seq(-1,-20,-1))){
+  approx <- forward.approx(sin, pi, h)
+  error <- abs(cos(pi)-approx)
+  forward.approx.table <- rbind(forward.approx.table, c(h, approx, error)) 
+}
+
+plot(abs(log(forward.approx.table[,"h"],10)), forward.approx.table[,"error"],
+     xlab="h [10^-x]", ylab="error (logscale)", type="o", log="y")
+
+# Repeat with central difference approx
+
+## 2.
+# a) Runge-Kutta
+
+# b) Numerically solve the decay ode and compare to Euler and RK error
+
+## 3. Use library function ode45 to solve the decay numerically
+
+## 4. Solve the predator-prey model numerically
+
+# a) k1=0.01, k2=0.1, k3=0.001, k4=0.05, prey(0)=50, pred(0)=15. t=0 -> 200
+
+# plot
+
+# b) Use Euler and RK to solve with h=10
+
+# plot comparing Prey solutions to ode45
+
+# c) Use k3=0.02
+
+## 5. Solve SIR model numerically from t=0 -> 20
+
+# a) a=0.5, b=1, S(0)=0.9, I(0)=0.1, R(0)=0
+
+# plot
+
+# b) a=3
+
+## 6. Decomp of dinitrogen pentoxygen into nitrogen dioxide and molecular oxygen
+
+# a) k1=1.0, k2=0.5, k3=0.2,k4=1.5, [N2O5]o=1, all other IC's=0, t=0 -> 10
+
+# b) Increase k4 to make the intermediate species -> 0