%%Matlab code for Improved Euler and Ralston RK2 method
clear all
close all
%Function for which solution have to do
f=@(t,y) y.*(sin(t)).^3;
%Improved Euler method
h=0.1;
% step size
t=0;
% initial t
y=1;
% initial y
t_eval=3; % at what point
we have to evaluate
n=(t_eval-t)/h; % Number of steps
t_euler(1)=t;
y_euler(1)=y;
for i=1:n
%improved Euler steps
m1=double(f(t,y));
m2=double(f((t+h),(y+h*m1)));
y=y+double(h*((m1+m2)/2));
t=t+h;
y_euler(i+1)=y;
t_euler(i+1)=t;
end
fprintf('\n\tThe solution using improved Euler
Method for h=%.2f at x(%.1f) is
%f\n',h,t_euler(end),y_euler(end))
%RK4 method
t=0;
% initial t
y=1;
% initial y
t_eval=3; % at what point
we have to evaluate
n=(t_eval-t)/h; % Number of steps
t_rk(1)=t;
y_rk(1)=y;
for i=1:n
%RK4 Steps
k1=h*double(f(t,y));
k2=h*double(f((t+(2*h/3)),(y+(2*k1/3))));
dy=(1/4)*(k1+3*k2);
t=t+h;
y=y+dy;
t_rk(i+1)=t;
y_rk(i+1)=y;
end
fprintf('\n\tThe solution using Rlaston Runge
Kutta 2 for h=%.2f at x(%.1f) is %f\n',h,t_rk(end),y_rk(end))
%%Plotting solution using Euler method
figure(1)
hold on
plot(t_euler,y_euler,'Linewidth',2)
plot(t_rk,y_rk,'Linewidth',2)
xlabel('t')
ylabel('y(t)')
title('Solution plot y(t) vs. t')
legend('Euler Heun Method','Ralston RK2
Method','Location','northwest')
grid on
%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%
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