Question

y'=t, y0)=1, solution: y(t)=1+t/2 y' = 2(1 +1)y, y(0)=1, solution: y(t) = +24 v=5"y, y(0)=1, solution: y(t) = { y'=+/yº, y(0)=1, solution: y(t) = (31/4+1)1/3

use Matlab

Answer 1

MATLAB Code:

close all
clear
clc

k = 0:1:5;
H = 0.1 * 2.^(-k);

f = @(t,y) t; % Given ODE
y_exact = @(t) 1 + t^2 / 2;
y(1) = 1; % Initial condition
ti = 0; tf = 1; % Interval of t
yf_vec = []; % Vector for storing y(t = 1) for different step sizes
for i = 1:length(H)
h = H(i);
t = ti:h:tf;
for j = 1:length(t)-1
k1 = f(t(j), y(j));
k2 = f(t(j) + 0.5*h, y(j) + 0.5*h*k1);
k3 = f(t(j) + 0.5*h, y(j) + 0.5*h*k2);
k4 = f(t(j) + h, y(j) + k3*h);
y(j + 1) = y(j) + (1/6)*(k1 + 2*k2 + 2*k3 + k4)*h; % RK4 Update
end
yf_vec = [yf_vec, y(end)]; % Store the value of y(t = 1)
end
figure, loglog(H, abs(y_exact(1) - yf_vec), 'o-'), xlabel('h'), ylabel('Error')
title(sprintf('Log-Log plot of Error of RK4 Method at t = 1\nODE: y'' = t'))

f = @(t,y) 2*(t + 1)*y; % Given ODE
y_exact = @(t) exp(t^2 + 2*t);
y(1) = 1; % Initial condition
ti = 0; tf = 1; % Interval of t
yf_vec = []; % Vector for storing y(t = 1) for different step sizes
for i = 1:length(H)
h = H(i);
t = ti:h:tf;
for j = 1:length(t)-1
k1 = f(t(j), y(j));
k2 = f(t(j) + 0.5*h, y(j) + 0.5*h*k1);
k3 = f(t(j) + 0.5*h, y(j) + 0.5*h*k2);
k4 = f(t(j) + h, y(j) + k3*h);
y(j + 1) = y(j) + (1/6)*(k1 + 2*k2 + 2*k3 + k4)*h; % RK4 Update
end
yf_vec = [yf_vec, y(end)]; % Store the value of y(t = 1)
end
figure, loglog(H, abs(y_exact(1) - yf_vec), 'o-'), xlabel('h'), ylabel('Error')
title(sprintf('Log-Log plot of Error of RK4 Method at t = 1\nODE: y'' = 2(t + 1)y'))

f = @(t,y) 5 * t^4 * y; % Given ODE
y_exact = @(t) exp(t^5);
y(1) = 1; % Initial condition
ti = 0; tf = 1; % Interval of t
yf_vec = []; % Vector for storing y(t = 1) for different step sizes
for i = 1:length(H)
h = H(i);
t = ti:h:tf;
for j = 1:length(t)-1
k1 = f(t(j), y(j));
k2 = f(t(j) + 0.5*h, y(j) + 0.5*h*k1);
k3 = f(t(j) + 0.5*h, y(j) + 0.5*h*k2);
k4 = f(t(j) + h, y(j) + k3*h);
y(j + 1) = y(j) + (1/6)*(k1 + 2*k2 + 2*k3 + k4)*h; % RK4 Update
end
yf_vec = [yf_vec, y(end)]; % Store the value of y(t = 1)
end
figure, loglog(H, abs(y_exact(1) - yf_vec), 'o-'), xlabel('h'), ylabel('Error')
title(sprintf('Log-Log plot of Error of RK4 Method at t = 1\nODE: y'' = 5t^4y'))

f = @(t,y) t^3 / y^2; % Given ODE
y_exact = @(t) (3 * t^4 / 4 + 1)^(1/3);
y(1) = 1; % Initial condition
ti = 0; tf = 1; % Interval of t
yf_vec = []; % Vector for storing y(t = 1) for different step sizes
for i = 1:length(H)
h = H(i);
t = ti:h:tf;
for j = 1:length(t)-1
k1 = f(t(j), y(j));
k2 = f(t(j) + 0.5*h, y(j) + 0.5*h*k1);
k3 = f(t(j) + 0.5*h, y(j) + 0.5*h*k2);
k4 = f(t(j) + h, y(j) + k3*h);
y(j + 1) = y(j) + (1/6)*(k1 + 2*k2 + 2*k3 + k4)*h; % RK4 Update
end
yf_vec = [yf_vec, y(end)]; % Store the value of y(t = 1)
end
figure, loglog(H, abs(y_exact(1) - yf_vec), 'o-'), xlabel('h'), ylabel('Error')
title(sprintf('Log-Log plot of Error of RK4 Method at t = 1\nODE: y'' = t^3/y^2'))

Plots:

Log-Log plot of Error of RK4 Method at t = 1 ODE: y'=t * 10-16 Error 10-2 10

Log-Log plot of Error of RK4 Method at t = 1 ODE: y' = 2(t+1)y Error 10-2 10

Log-Log plot of Error of RK4 Method at t = 1 ODE: y' = 5ty Error 10-2 10

Log-Log plot of Error of RK4 Method at t = 1 ODE: y'=tly2 0 Error 10-2