# Example Euler method for y(t) = c * e^t
# Given: y'(t) = f(t,y) = y(t), Initial value: y(0) = c

# solve by y_n+1 = y_n + h * y_n

c <- 1 # Initial value
n = 10
h = 1 # Change for better approximation
t <- seq(0, n, by = h)
y_orig <- c * exp(t)

y_approx <- numeric(length(t))
y_approx[1] <- c
i <- 2
for (t_i in t[-1])
{
  y_approx[i] <- y_approx[i-1] + h * y_approx[i-1]
  i = i + 1
}

library(ggplot2)
source("defines.R")
ggplot() + plot_style +
  geom_line(aes(x = t, y = y_orig, color="Function"), size = 1) + 
  geom_line(aes(x = t, y = y_approx, color="Approx h=1"), size = 1) +
  scale_color_manual(name = "Values", values = c("Function"="blue", "Approx h=1"="red"))

# try different values for h
c <- 1 # Initial value
n = 5
hs <- c(1, 0.5, 0.1, 0.01, 0.001)
t <- seq(0, n, by = last(hs))
y <- c * exp(t)

df <- data.frame("Name" = "y", "Time" = t, "Value" = y)

for (h in hs)
{ # Euler method for every h
  y <- numeric()
  y[1] <- c
  t <- numeric()
  t[1] <- 0
  i <- 2
  while((t[i-1] + h) <= n)
  {
    t[i] <- t[i-1] + h
    y[i] <- y[i-1] + h * y[i-1]
    i = i + 1
  }
  df <- rbind(df, data.frame("Name" = paste0("h=",h), "Time" = t, "Value" = y))
}
ggplot(df, aes(x = Time, y = Value, color = Name)) + geom_line() + plot_style