Question

1 st s2, y(1)1 The exact solution is given by yo) - = . 1+Int Write a MATLAB code to approximate the solution of the IVP using Midpoint (RK2) and Modified Euler methods when h [0.5 0.1 0.0s 0.01 0.005 0.001]. A) Find the vector w mid and w mod that approximates the solution of the IVP for different values of h. B) Plot the step-size h versus the relative error of both in the same figure using the LOGLOG function. . You use this: loglog (h,RE mid, +,h, RE mod,-o') legend ( 'Midpoint-RK2, 'Modified Euler') xlabel ('h') ylabel ( 'Relative Error') C) Plot the step-size h versus the computational times of both in the same figure using the LOGLOG function. 2-(Bonus) Repeat the same but using Runge-Kutta of order four (R K4).

Answer 1

Matlab code for 2nd order Runge Kutta and Modified Euler clear all close all %function for solving ODE using rk2 %function for exact solution fun ext-e(t,y t./(1+log (t)) tall step size hh-10.5 0.1 0.05 0.01 0.005 0.0011 for ii-1length( hh) t(1)=1;yA( 1)=1; t in-t (1); t_max-2; %Rung %initial %Initial t Final t conditions e Kutta 2 iterations number of steps n-(t_max-t in) /h; tic; for i-1:n k1-h fun (t(i),ya(i)) t (i+1)-t_inti*h; yA(i+1)-double (yA(i)tk2); end time, mid ( ii )-toc ; plotting the solution using RK2 figure(1) hold on plot (t,yA) legendinfo(11)-sprintf("RK2 for step RE_mid(ii)-norm(yA-fun_ext(t)) length %2.3f',h); clear t; clear yA; xlabel('t ylabel('Y(t) title( 'y (t) vs t plot using RK2 method) legend (legendinfo, location', "northwest' for ii-1length ( hh) ун-1; %initial conditions t_in-l; t_max-2; %Modǐfled Euler step size hshh ( ii); %Initial t Final t

number of steps n-(t_max-t_in)/h; УЗ (1 )-yE; tici for i-l:n improved Eular steps ml-double (fun(t,yH)); m2-double (fun ( (t+h), (yH+h*m1))) yH-yH+double(h* ((ml+m2)/2) t-t+h; y3 (i+)-yH end time-mod ( i i ) =toc ; %plotting the solution using Modified Euler figure (2) hold on plot (t3,y3) legendinfo(11)-sprintf('improved Euler for step length %2.3f',h); RE mod (ii)-norm(y3-fun_ext (t3)) clear t3; clear y3; xlabel('t' title( 'y(t) vs t plot using improved Euler method') end ylabel('Y(t) legend (legendinfo, location',northwest') figure (3) hold on loglog (hh, REmid, "-+ ',hh, RE_mod,"-o) legend'Midpoint-RK2, 'Modified Euler') xlabel('h') ylabel Relative error') title( 'Loglog plot of stepsize vs. rel error') figure(4) hold or loglog (hh, time mid,"-,hh,time mod, -o') legend Midpoint-RK2, 'Modified Euler') xlabel'h') ylabel('time') title( 'Loglog plot of stepsize vs. time' for ii-1length( hh) t-1; R-I; %initial conditions t in-1 t_max-2; 8Modified Euler step size %Initial t %Final t %number of steps

n- (t max-t in)/h; tic; for i-l:n %RK4 STEPS kl-h*double (fun (t, yR)); k2-h double (fun ((tth/2) , (yR+kl/2))); k3-h double (fun ( (tth/2, (yR+k2/2))) k4-h*double (fun ((t+h), (yR+k3))) dy-(1/6) (k1+2* k2+2*k3+k4) t-t+h; timerk4 ( ii )-toc ; %plotting the solution using Modified Euler figure (5) hold on plot (t4,y4) legend info(11)-sprintf('rk 4 solution for step length RE_rk4 (ii)-norm(y4-fun_ext (t4)) clear t4; clear y4; - %2.3f',h); end xlabel ( ' t' ) ylabel(Y (t)' title( 'y (t) vs t plot using rk4 method') legend (legendinfo, 'location, 'northwest')

yt) vs t plot using RK2 method 1.2 RK2 for step length 0.500 RK2 forstep length 0.100 1.18 RK2 for step length O.050 -RK2 for step length 0.010 -RO for step length 0.006 0.001 1.16 1.14 1.12 1.1 1.08 1.06 104 1.02 1.2 1.3 1.5 1.6 1.8 1.9 tt) vs t plot using improved Euler method Improved Euler for step length 0.500 mproved Euler for step length O.100 ·Improved Eler tr step length 0.050 Improved Euler for step length O.010 Improved Euler for step length 0.005 0.001 1.18 1.16 ed Euler for 1.14 1.12 1.1 1.08 1.06 1.04 1.02 1.2 1.3 14 1.5 1.8 1.9

Loglog plot of stepsize vs. rel error 0.02 Midpoint RK2 Modified Euler 0018 0016 0014 0012 001 0.008 0.006 0.004 0.002 0.05 0.1 0.15 02 025 0.3 035 0A 0.45 05 x 10 Loglog plot of stepsize vs. time Midpoint RK2 Modified Euler 4.5 3.5 2.5 0.5 0.05 0.1 0.15 02 025 0.3 035 04 045 05

ytt) vs t plot using rk4 method 1.2 rk4 solution for step length 0.500 rk4 solution for step length 0.100 rk4 solution for step length 0.050 rk4 solution for step length 0.010 -ried solution for step length 0.005 0.001 1.18 1.16 rk4 solution for 1.14 1.12 1.1 1.08 1.06 104 .02 1 13 14 15 16118 92 Published with MATLAB R2018a

%%Matlab code for 2nd order Runge Kutta and Modified Euler
clear all
close all
%function for solving ODE using rk2
fun=@(t,y) (y./t)-(y./t).^2;
%function for exact solution
fun_ext=@(t,y) t./(1+log(t));
%all step size
hh=[0.5 0.1 0.05 0.01 0.005 0.001];

for ii=1:length(hh)
  
    t(1)=1;yA(1)=1;   %initial conditions
    t_in=t(1);        %Initial t
    t_max=2;          %Final t
    %Runge Kutta 2 iterations
    h=hh(ii);
    %number of steps
    n=(t_max-t_in)/h;
    tic;
    for i=1:n
     
        k1=h*fun(t(i),yA(i));
        k2=h*fun(t(i)+(1/2)*h,yA(i)+(1/2)*k1);
      
        t(i+1)=t_in+i*h;
        yA(i+1)=double(yA(i)+k2);
    end
    time_mid(ii)=toc;
    %plotting the solution using RK2
    figure(1)
    hold on
    plot(t,yA)
    legendinfo{ii}=sprintf('RK2 for step length %2.3f',h);
    RE_mid(ii)=norm(yA-fun_ext(t));

    clear t; clear yA;
end
    xlabel('t')
    ylabel('Y(t)')
    title('y(t) vs t plot using RK2 method')
    legend(legendinfo,'location','northwest')
  

for ii=1:length(hh)
  
    t=1;
    yH=1;   %initial conditions
    t_in=1;        %Initial t
    t_max=2;       %Final t
    %Modified Euler step size
    h=hh(ii);
    %number of steps
    n=(t_max-t_in)/h;
    t3(1)=1;
    y3(1)=yH;
    tic;
    for i=1:n
    %improved Eular steps
       m1=double(fun(t,yH));
       m2=double(fun((t+h),(yH+h*m1)));
       yH=yH+double(h*((m1+m2)/2));
       t=t+h;
       y3(i+1)=yH;
       t3(i+1)=t;
    end
    time_mod(ii)=toc;
    %plotting the solution using Modified Euler
    figure(2)
    hold on
    plot(t3,y3)
    legendinfo{ii}=sprintf('Improved Euler for step length %2.3f',h);
    RE_mod(ii)=norm(y3-fun_ext(t3));

    clear t3; clear y3;
end
    xlabel('t')
    ylabel('Y(t)')
    title('y(t) vs t plot using improved Euler method')
    legend(legendinfo,'location','northwest')

    figure(3)
hold on
loglog(hh,RE_mid,'-+',hh,RE_mod,'-o')
legend('Midpoint=RK2','Modified Euler')
xlabel('h')
ylabel('Relative error')
title('Loglog plot of stepsize vs. rel error')

figure(4)
hold on
loglog(hh,time_mid,'-+',hh,time_mod,'-o')
legend('Midpoint=RK2','Modified Euler')
xlabel('h')
ylabel('time')
title('Loglog plot of stepsize vs. time')

for ii=1:length(hh)
  
    t=1;
    yR=1;   %initial conditions
    t_in=1;        %Initial t
    t_max=2;       %Final t
    %Modified Euler step size
    h=hh(ii);
    %number of steps
    n=(t_max-t_in)/h;
    t4(1)=1;
    y4(1)=yR;
    tic;

    for i=1:n
    %RK4 STEPS
       k1=h*double(fun(t,yR));
       k2=h*double(fun((t+h/2),(yR+k1/2)));
       k3=h*double(fun((t+h/2),(yR+k2/2)));
       k4=h*double(fun((t+h),(yR+k3)));
       dy=(1/6)*(k1+2*k2+2*k3+k4);
       t=t+h;
       yR=yR+dy;
       t4(i+1)=t;
       y4(i+1)=yR;

    end
    time_rk4(ii)=toc;
    %plotting the solution using Modified Euler
    figure(5)
    hold on
    plot(t4,y4)
    legendinfo{ii}=sprintf('rk4 solution for step length %2.3f',h);
    RE_rk4(ii)=norm(y4-fun_ext(t4));
    clear t4; clear y4;
end

    xlabel('t')
    ylabel('Y(t)')
    title('y(t) vs t plot using rk4 method')
    legend(legendinfo,'location','northwest')
%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%