Cosas de la práctica N° 1


## ejercicio 2

f <- function(t,x){
  return ((x-5)*((cos(t))^2-0.5))
}


n <- 10/0.01
h <- 0.01


t <- 0
xeul <- 0

for(i in c(1:n)){
  t[i+1] <- t[i]+h
  xeul[i+1] <- xeul[i] + h * f(t[i],xeul[i])
}

plot(t,xeul,ylim=c(-2,10),col="green",type="l")

for (k in c(1:10)) {
  t[1] <- 0
  xeul[1] <- k
  
  for(i in c(1:n)){
    t[i+1] <- t[i]+h
    xeul[i+1] <- xeul[i] + h* f(t[i],xeul[i])
  }
  
  lines(t,xeul,col="blue")
}


##########################################################

## ejercicio 4

euler1 <- function(f,t0,tf,h,y0) {
  n <- (tf-t0)/h
  t <- t0
  y <- y0
  
  for(i in 1:n) {
    y0 <- y0 + h * f(t0, y0)
    t0 <- t0 + h
    t <- c(t, t0)
    y <- c(y, y0)
  }
  
  return(data.frame(t = t, y = y))
}  
t <- 0
y <- 0

euler2 <-function(f,t0,tf,h,y0){
  n<-(tf-t0)/h
  t[1]<-t0
  y[1]<-y0
  i=1
  while(i<=n){
    t[i+1]<-t[i]+h
    y[i+1]<-y[i]+h*f(t[i],y[i])
    i<-i+1
  }
  return(data.frame(t,y))
}



#Ejemplo 1
f1<-function(t,y){
  return((y-5)*((cos(t))^2-0.5))
}

e1<-euler1(f1,0,10,0.25,0)
show(e1)

plot(e1[,1],e1[,2],ylim=c(-5,5),type = "l", col="red")



e2<-euler2(f1,0,10,0.25,0)
show(e2)

plot(e2[,1],e1[,2],ylim=c(-5,5),type = "l", col="green")


####################################################

## ejercicio 6

f<-function(t,y,k){
  return(k*y)
}

t <- 0
y <- 0

euler <-function(f,t0,tf,h,y0,k){
  n<-(tf-t0)/h
  t[1]<-t0
  y[1]<-y0
  i=1
  while(i<=n){
    t[i+1]<-t[i]+h
    y[i+1]<-y[i]+h*f(t[i],y[i],k)
    i<-i+1
  }
  return(data.frame(t,y))
}

eulerimp <-function(t0,tf,h,y0,k){
  n<-(tf-t0)/h
  t[1]<-t0
  y[1]<-y0
  i=1
  while(i<=n){
    t[i+1]<-t[i]+h
    y[i+1]<-(1/(1-h*k))*y[i]
    i<-i+1
  }
  return(data.frame(t,y))
}

e1<-euler(f,0,1,0.01,1,1)
e1otro<-eulerimp(0,1,0.01,1,1)
plot(e1[,1],e1[,2],ylim=c(1,3),type = "l", col="red")
lines(e1otro[,1],e1otro[,2], col="green")
curve(exp,0,1,n=1001,col="blue",add=TRUE)


e10<-euler(f,0,1,0.01,1,10)
e10otro<-eulerimp(0,1,0.01,1,10)
plot(e10[,1],e10[,2],ylim=c(0,2000),type = "l", col="red")
curve(exp(10*x),0,1,n=1001,col="blue",add=TRUE)
lines(e10otro[,1],e10otro[,2], col="green")

#############################################################

## ejercicio 8

# limpiar variables

rm(list=ls())

# datos iniciales

t0 <- 0
x0 <- 0.5

# función de la EDO

f <- function(t,x){
  x^2
}

# tiempo final de la aproximación

T <- 2.1

# número de pasos y tamaño del paso

n <- 10
h <- (T-t0)/n


# método de Euler
t <- t0
xeul <- x0

for(i in c(1:n)){
  t[i+1] <- t[i]+h
  xeul[i+1] <- xeul[i] + h* f(t[i],xeul[i])
}


# método de Euler con paso adaptado

lambda <- 0.1
hmin <- 0.001

tad <- t0
xeulad <- x0
c <- f(t0,x0)
i <- 1

while(lambda/hmin >= c[i] )
{
  xeulad[i+1]<- xeulad[i]+lambda
  tad[i+1] <- tad[i]+lambda/f(tad[i],xeulad[i])
  c[i+1] <- f(tad[i+1],xeulad[i+1])
  i <- i+1
}

# graficar

plot(t,xeul, ylim=c(0,200),type = "l", col="red")
lines(tad,xeulad,col="green")
curve(1/(2-x),0,2,n=100001,col="blue",add=TRUE)