Question

In Exercise, use the Runge-Kutta method with the given number n of steps to approximate the solution to the initial-value problem specified. Your answer should include a table of approximate values of the dependent variable. It should also include a sketch of the graph of the approximate solution. Compare the graphs that you get from the Runge-Kutta method to those that come from Euler's method and improved Euler's method. If your computer has a built-in routine for the numerical solution of differential equations, compare these graphs with those that come from the built-in numerical solver. ax = 2/9, y0 = 2, Ost<2, n=4

Answer 1

MATLAB Script:

close all
clear
clc

f = @(t,y) exp(2/y); % Given ODE
t0 = 0; tf = 2; % Intervals of t
y0 = 2; % Initial condition
n = 4;
h = (tf - t0)/n; % Step Size
t = t0:h:tf;

[t_e, y_e] = ode45(f, [t0, tf], y0); % Exact solution
plot(t_e, y_e), hold on % Plot the exact solution

y = my_euler(t0, y0, tf, h, f);
plot(t, y, 'o-') % Plot Euler's method solution

y = my_imp_euler(t0, y0, tf, h, f);
plot(t, y, '*-') % Plot Improved Euler's method solution

y = my_rk4(t0, y0, tf, h, f);
plot(t, y, '^-'), hold off % Plot RK4 method solution
fprintf('RK4 Method Solution\n-----------------------------------------------------------\n')
fprintf('%-10s%-20s\n', 't(i)', 'y(i)')
for i = 1:length(t)
fprintf('%-10.4f%-20.4f\n', t(i), y(i))
end

xlabel('t'), ylabel('y(t)')
legend('Exact Solution', 'Euler''s Method', 'Improved Euler''s Method', 'RK4 Method', 'Location', 'northwest')
title('Solution of ODE')

function y = my_euler(t0, y0, tf, h, f)
y(1) = y0;
t = t0:h:tf;
for i = 1:length(t)-1
y(i+1) = y(i) + h*f(t(i), y(i)); % Euler Update
end
end

function y = my_imp_euler(t0, y0, tf, h, f)
y(1) = y0;
t = t0:h:tf;
for i = 1:length(t)-1
f1 = f(t(i), y(i));
f2 = f(t(i) + h, y(i) + h*f1);
y(i+1) = y(i) + (h/2)*(f1 + f2); % Improved Euler's Update
end
end

function y = my_rk4(t0, y0, tf, h, f)
y(1) = y0;
t = t0:h:tf;
for i = 1:length(t)-1
k1 = f(t(i), y(i));
k2 = f(t(i) + 0.5*h, y(i) + 0.5*h*k1);
k3 = f(t(i) + 0.5*h, y(i) + 0.5*h*k2);
k4 = f(t(i) + h, y(i) + k3*h);
y(i + 1) = y(i) + (1/6)*(k1 + 2*k2 + 2*k3 + k4)*h; % RK4 Update
end
end

Output:

Command Window RK4 Method Solution t(i) У(i) 0.0000 2.0000 0.5000 3.1046 1.0000 3.9855 4.7755 1.5000 2.0000 5.5135 f > |

Plot:

Solution of ODE Exact Solution Euler's Method -Improved Euler's Method A-RK4 Method 5.5 5 4.5 3.5 3 2.5 0.6 0.8 1.2 1.4 1.8 0.2 0.4 1.6