Question

SOLVE USING MATLAB ONLY AND SHOW FULL CODE. PLEASE TO SHOW TEXT BOOK SOLUTION. SOLVE PART D ONLY

Apply Euler's Method with step sizes h # 0.1 and h 0.01 to the initial value problems in Exercise 1. Plot the approximate solutions and the correct solution on [O, 1], and find the global truncation error at t-1. Is the reduction in error for h -0.01 consistent with the order of Euler's Method? REFERENCE: Apply the Euler's Method with step size h #: 1 /4 to the initial value problem on [0,1]. (a)vo (0) = 0 1: = 2n + 2)2 n(o)5 92 (0) 0 Find the global truncation errors ofyi and y2 at t = 1 by comparing with the correct solutions (a) yi (t)-d cost.)2(t)--e, sin t (b) y(t) e", cost, y2(,) # e-r sin r

Answer 1

%%Matlab code for Euler's forward clear all close all 888888,,88,,88움움8888888888움움88888888888888888888888888888888888888 %function for Euler equation solution fl-e(yl,y2,t) yl+3*y2; for the functions yl ext- (t) 3*exp(-t)+2*exp (4*t); y2 ext- (t) -2*exp (-t)+2*exp (4*t) displaying the functions fprintf 'For functions \n') disp (f1) disp(f2) conditions %yl ( 0 )-1 %initial yl-in-5; %initial t t_in-0; %t end values t end-1; h=0.1; %step length results of h-0.1 [x1_result,x2_result,t_result]-Euler (f1, £2,ylin,y2_in,t_in, t_end, h); error finding err1-abs (yl ext(t end)-x1 result (end) err2-abs (y2_ext(t_end) -x2_result (end)); figure (1) hold on plot (t_result,x1_result) xlabel('t') ylabel('yl(t) title('t vs. y1(t) plot' figure (2) hold on plot(t_result, x2_result) xlabel(t' ylabel ('y2(t)' title( 't vs. y2(t) plot' fprint f ("AtGlobal truncation error for y1 for h-%f at t-%f 1s %f. n',h,t end, errl) fprint f ( '\tGlobal truncation error for y1 for h=%f at t=%f 1s %f.\n n',h,t end, err2)

h-0.01; %step length results of h-0.1 [x1_result,x2_result,t_result]-Euler (f1, £2,y1 in,y2_in,t_in, t end, h); figure (1) plot (t_result,xl_result) plot (t_result,y1 ext(t_result)) legend('Solution for h-0.1', Solution for h-0.01', 'Exact solution) figure (2) hold on plot (t_result, x2_result) plot (t_result,y2 ext(t_result)) legend ('Solution for h=0.1', 'solution for h=0.01', 'Exact solution') terror finding er r 1-abs (y l_ext ( tend)-X1-result ( end ) ) ; err2-abs (y2一ext ( t-end)-x2-result ( end ) ) ; fprint f ("AtGlobal truncation error for y1 for h-%f at t-%f 1s %f. n',h,t end, errl) fprint f ("AtGlobal truncation error for y1 for h=%f at t=%f 1s %f.\n n',h,t end, err2) function [x1_result, x2_result,t_result]-Euler(fl,f2,yl_in,y2_in,t_in,t_end, h) in: ht end tn=t Euler steps x1_result (1)-yl in; x2_result(1)-y2_in; t_result (1)-t_in; %loop for Euler step for finding the solution for i-l:length (tn)-1 t_result (i+1)- t_result(i)+h; x1 result(i +1 )-xl-result ( İ) +h * double ( fl (x1-result (i) , x2-result(1), t-result(1)) ) ; x2 result(i +1 ).x2_result ( İ) +h * double ( f2 (x1-result (i) , x2-result ( i) ,t-result(1)) ) ; end end 용욤욤응용욤욤응용욤욤응용음음용 End of Code 응응용욤응응용욤응응욤욤응응욤욤응응용

For functions e(yl,y2,t)yl+3*y2 Global truncation error for yl for h-0.100000 at t-1.000000 is 51.402972. Global truncation error for yl for h-0.100000 at t-1.000000 is 51.306967. Global truncation error for y1 for h=0.010000 at t= 1.000000 is 8.191945. Global truncation error for y1 for h=0.010000 at t-1.000000 is 8. 182709. tvs. y1 t) plot 120 Solution for h-0.1 Solution for h 0.01 Exact solution 100 80 40 20 0.1 0.2 0 0.3 04 0.5 0.6 0.7 0.8 0.9

tvs. y2tt) plot 120 Solution for h:0.1 Solution for h-0.01 Exact solution 100 80 60 40 20 0 .1 0 03 04 05 0608 0.9 Published with MATLAB R2018a

%%Matlab code for Euler's forward
clear all
close all

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

%function for Euler equation solution
f1=@(y1,y2,t) y1+3*y2;
f2=@(y1,y2,t) 2*y1+2*y2;

%for the functions
y1_ext=@(t) 3*exp(-t)+2*exp(4*t);
y2_ext=@(t) -2*exp(-t)+2*exp(4*t);

%displaying the functions
fprintf('For functions \n')
disp(f1)
disp(f2)

%Initial conditions
y1_in=5;   %y1(0)=1
y2_in=0;   %y2(0)=0
%initial t
t_in=0;
%t end values
t_end=1;
h=0.1; %step length

%results of h=0.1
[x1_result,x2_result,t_result]=Euler(f1,f2,y1_in,y2_in,t_in,t_end,h);
%error finding
err1=abs(y1_ext(t_end)-x1_result(end));
err2=abs(y2_ext(t_end)-x2_result(end));

figure(1)
hold on
plot(t_result,x1_result)
xlabel('t')
ylabel('y1(t)')
title('t vs. y1(t) plot')

figure(2)
hold on
plot(t_result,x2_result)
xlabel('t')
ylabel('y2(t)')
title('t vs. y2(t) plot')

fprintf('\tGlobal truncation error for y1 for h=%f at t=%f is %f.\n',h,t_end,err1)
fprintf('\tGlobal truncation error for y1 for h=%f at t=%f is %f.\n\n',h,t_end,err2)

h=0.01; %step length

%results of h=0.1
[x1_result,x2_result,t_result]=Euler(f1,f2,y1_in,y2_in,t_in,t_end,h);
figure(1)

plot(t_result,x1_result)
plot(t_result,y1_ext(t_result))
legend('Solution for h=0.1','Solution for h=0.01','Exact solution')

figure(2)
hold on
plot(t_result,x2_result)
plot(t_result,y2_ext(t_result))
legend('Solution for h=0.1','Solution for h=0.01','Exact solution')

%error finding
err1=abs(y1_ext(t_end)-x1_result(end));
err2=abs(y2_ext(t_end)-x2_result(end));

fprintf('\tGlobal truncation error for y1 for h=%f at t=%f is %f.\n',h,t_end,err1)
fprintf('\tGlobal truncation error for y1 for h=%f at t=%f is %f.\n\n',h,t_end,err2)

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

function [x1_result,x2_result,t_result]=Euler(f1,f2,y1_in,y2_in,t_in,t_end,h)

    tn=t_in:h:t_end;
    % Euler steps
    x1_result(1)=y1_in;
    x2_result(1)=y2_in;
    t_result(1)=t_in;

    %loop for Euler step for finding the solution
  
    for i=1:length(tn)-1

        t_result(i+1)= t_result(i)+h;
        x1_result(i+1)=x1_result(i)+h*double(f1(x1_result(i),x2_result(i),t_result(i)));
        x2_result(i+1)=x2_result(i)+h*double(f2(x1_result(i),x2_result(i),t_result(i)));
     
    end
  
end

                %%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%%