Question

5. Consider the following second order IVP y2y te - t, 0 t1 y(0)/(0) 0 = ( y(t). Transform the above IVP to system of first order (a) Let u(t)y(t) and u2(t) IVP of u and u2. (b) Find y(t) by solving the system with h 0.1 (c) Compare the results to the actual solution y(t) = %et - te 2e t - 2.

Answer 1

5. -t te (02,5 Let dy dt dt duy 2. dtr P. dy Tuitiol ondition ui o O 2- dt uing Euler metho d NO -21 dt -tet2et-t2 3.t Lo

Matlab code for system of ODE using Euler's forward clear all close all &All Parameters 8functions for Euler equation solution ul2 12-ul+t*exp (t)-t; all step size h 0.1 Initial values u10 0 u20-0; t0 0 t end values tend-1; tn t0:h:tend 8 Euler steps ul_result(1)=u10; u2_result(1)=u20 t result (1=t0 ; Sexact solution y_ext e (t) (1/6).*t.^3.*exp (t)-t.*exp (t)+2.*exp(t)-t-2; fprintft y=ul(0) 8 f\n' ,ul_result (1)) fprintf\t y_ext= %f\n\n' ,y_ext (t_result (1)) ) for i-1:length (tn)-1 t_result (i+1) ul resulti ul_result (i) +h*double(f(ul_result (i),u2_result(i),t_result(i)) ); t_result (i) +h; +1 u2 result(i +1u2_result ( i) +h*double (g(ul_result (i) ,u2_result(i),t_result(i))); fprintfAfter %d iteration \n',i) fprintf( fprintf\t y_ext= %f\n\n',y_ext (t_result(i+1)) \t y=ul(%2.2f)= %f\n',t_result(i+1),ul_result (i+1)) end %plotting the solution hold on plot (t_result , ul_result) plot (t result, y_ext(t_result)) xlabelt' ylabel'y(t) ie Y(t) vs. t plot') legendEuler solution', 'Exact Solution') box on grid on $88888888888888*** End of Code %%$88888888*$888 1

y-ul (0)- 0.000000 yext 0.000000 After 1 iteration y=u1 (0.10) = 0.000000 y ext 0.000009 After 2 iteration 000000 yul (0 yext 0.000154 After 3 iteration yul (0.30) = 0.000105 y ext 0.000834 After 4 iteration yul (0.40) = 0.000674 y ext 0.002832 After 5 iteration yul (0.50)= 0.002406 yext 0.007430 After 6 iteration yul (0.60) = 0.006444 y ext 0.016563 After 7 iteration y-ul (0.70) = 0.014509 yext 0.032998 After 8 iteration y=ul (0.80)= 0.029056 yext 0.060562 After 9 iteration y=ul (0.90)= 0.053463 yext 0.104405 After 10 iteration y=ul (.00)0.092265 yext 0.171329 2

y(t) vs. t plot 0.18 Euler solufon Exact Solution 0.16 0.14 0.12 0.1 0.08 0.06 004 0.02 0 n 1 02 03 0 4 0.5 0.6 0.9 t Published with MATLAB R2018a

%%Matlab code for system of ODE using Euler's forward
clear all
close all
%All Parameters

%functions for Euler equation solution
f=@(u1,u2,t) u2;
g=@(u1,u2,t) 2*u2-u1+t*exp(t)-t;
%all step size
h=0.1;
%Initial values
u10=0;
u20=0;
t0=0;

%t end values
tend=1;
tn=t0:h:tend;
% Euler steps

u1_result(1)=u10;
u2_result(1)=u20;
t_result(1)=t0;

%exact solution
y_ext=@(t) (1/6).*t.^3.*exp(t)-t.*exp(t)+2.*exp(t)-t-2;     
        fprintf('\t y=u1(0)= %f\n',u1_result(1))
        fprintf('\t y_ext= %f\n\n',y_ext(t_result(1)))
    for i=1:length(tn)-1

        t_result(i+1)= t_result(i)+h;
        u1_result(i+1)=u1_result(i)+h*double(f(u1_result(i),u2_result(i),t_result(i)));
        u2_result(i+1)=u2_result(i)+h*double(g(u1_result(i),u2_result(i),t_result(i)));
        fprintf('After %d iteration \n',i)
        fprintf('\t y=u1(%2.2f)= %f\n',t_result(i+1),u1_result(i+1))
        fprintf('\t y_ext= %f\n\n',y_ext(t_result(i+1)))
      
    end
    %plotting the solution
    hold on
    plot(t_result,u1_result)
    plot(t_result,y_ext(t_result))
  
    xlabel('t')
    ylabel('y(t)')
    title('y(t) vs. t plot')
    legend('Euler solution','Exact Solution')
    box on
    grid on
    %%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%