Question

25.5 Solve from 0 to 3 with h = 0.1 using (a) Heun (without corrector) and (b) Ralston's 2nd-order RK method: dy = y sin3 (1) y(0) = 1

Answer 1

Matlab code or Improved Euler and Ralston RK2 method clear all close all %Function for which solution have to do f-2(t,y) y-*(sin(t)).3; Improved Euler h-0.li t-0 y-l; t_eval-3; nm(t-eval-t)/h; t-euler ( 1 ) =t ; y_euler(1)-y; for i=1:n improved Euler method step size s initial t % initial y % at what point we have to evaluate % Number of steps steps ml-double(f (t,y)); m2-double(f ((t+h), (y+h*ml))); y=y+double ( h* ( ( m1 +m2 ) /2 ) ) ; t-t+hi y_euler(i+1)-y; teuler (1+1)=t; - end fprintf(n\tThe solution using improved Euler Method for h-8.2f x(8. lf) is %f\n',h,t_euler ( end ),y-euler(end)) at RK4 method t-0; y=1; t_eval-3 ns(t-eval-t)/h; t_rk ( 1 )=t; y rk(1)-y for i-1: n %RK4 Steps % initial t % initial y % at what point we have to evaluate % Number of steps kl-h*double (f (t,y) k2-h double (f ( (t+ (2wh/3)), (y+ (2*k1/3)))); dy-(1/4)*(kl+3*k2); t-t+h y=y +dy ; end fprintfntThe solution using Rlaston Runge Kutta 2 for h-.2f x(8. lf ) is %f\n',h,trk(end ),y-rk(end)) at 88Plotting 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 용号号응%号号응용号号%%号号%% End of Code 융용용융융욤욤응응용용융융욤욤 The solution using improved Euler Method for h-0.10 at x(3.0) įs 3.787545 The solution using Rlaston Runge Kutta 2 for h-0.10 at x(3.0) is 3.788044 Solution plot ylt) vs.t Euler Heun Method Ralston RK2 Method 3.5 巳2.5 1.5 35 0.5 1.5 2.5 Published with MATLAB R2018a

%%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 %%%%%%%%%%%%%%%