NewtonMin-12-02-2021, R

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# Gráfico em 3 dimensões

# Curva
# Superfície z=f(x,y)

x <- seq(0,2.5, by=0.05)
y <- seq(0,1,by=0.05)
mf<-function(s,r){z=sqrt(s*r)*exp(-sqrt(s^3)/2-r^2)# f(x,y); auxiliar para gráfico
z
}

require(grDevices) # for trans3d
z <- outer(x, y, mf)
z[is.na(z)] <- 4
op <- par(bg = "white")
persp(x, y, z, theta = 60, phi = 20, expand = 0.5)

# Curva (u(t),v(t))

t=seq(0,2.5,by=0.01)
u<-function(s){s*cos(s)+2}
v<-function(s){(-s^2)*sin(s)/4+1}

curve(u,0,2.5)

curve(v,0,2.5)

plot(u(t),v(t),'l',col="red")   # curva (u(t),v(t))

persp(x, y, z, theta = 60, phi = 20, expand = 0.5)->res
lines(trans3d(u(t),v(t),mf(u(t),v(t)), res), col = "red", lwd = 2) # Inclui curvas no gráfico em 3d

# Método de Euler oara resolver a equação e'(t)=grad f(e(t)), e(0)=(0.5,1)
hh=0.0001
gradmf<-function(u){          # aproximação de f'(x,y)
dfx=(mf(u[1]+hh,u[2])-mf(u[1]-hh,u[2]))/(2*hh)
dfy=(mf(u[1],u[2]+hh)-mf(u[1],u[2]-hh))/(2*hh)
c(dfx,dfy)
}

t0=0  # tempo inicial
tf=20  # t final 
e0=c(2.4,0.1) # condição inicial 
n=200
h=(tf-t0)/n  # Tamanho do passo

tt=seq(t0,tf,by=h)

Y=matrix(0,2,length(tt))
Y[,1]=e0

for ( i in 1:(length(tt)-1)){
Y[,i+1]=Y[,i]+h*gradmf(Y[,i])
}

Y[,length(tt)]
mf(Y[1,length(tt)],Y[2,length(tt)])

lines(trans3d(Y[1,],Y[2,],mf(Y[1,],Y[2,]), res), col = "blue", lwd = 2) # Inclui curvas

plot(t,mf(u(t),v(t)),'l',col="red")   # curva f(u(t),v(t))

plot(tt,mf(Y[1,],Y[2,]),col = "blue",'l')

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Absolute running time: 1.59 sec, cpu time: 0.94 sec, memory peak: 52 Mb, absolute service time: 1,8 sec