Need help with MATLAB code. Fuction file euler. m is given below. ..............

Question

Need help with MATLAB code.

Fuction file euler. m is given below.

........................................................................................................................................................................

Modify the M-file euler.m to implement the algorithm for Improved Euler. Call the new file impeuler.m (include the file in your report). Test your code for the ODE y'= 3y, y(0) = 2. First enter the function f(t, y) = 3y as anonymous function and then enter the following:

>> [t5,y5] = impeuler(f,[0,.5],2,5); % use @f if defined in separate function
>> [t5,y5]

ans = 0 2.0000
0.1000 2.6900
0.2000 3.6181
0.3000 4.8663
0.4000 6.5451
0.5000 8.8032

Compare your output with the one above. Note that the improved Euler approximation with 5 steps is already more accurate than the Euler approximation with 50 steps! (hence the “improved”)

Explanation / Answer

clc
clear all
%yn means value of y solved numerical methods
f=@(t,y) 3*y;
yn(1)= 2; %initial value matlab index reads from 1
t0=0;
tf=0.5;
h=0.1; %steplength
tspan=t0:h:tf;
N=tf/h+1;   % +1 is since t tvalue starts from 0

for n=1:N-1
  
    %improved Euler method %y(n+1)=y(n)+h/2*(f(t(i),y(i))+f(t(i)+h,y(i)+h*f(t(i),y(i)))
    yn(n+1)=yn(n)+h/2*(f(tspan(n),yn(n))+ 3*(yn(n)+h*f(tspan(n),yn(n))));
    %In the aboveline we have to write 3*y f(t(i)+h,y(i)+h*f(t(i),y(i))) in place
    %since yn matrix size & it changes for every iteration

end
t= tspan'
y = yn'    % change t and y from row to column vectors

output

t =

         0
    0.1000
    0.2000
    0.3000
    0.4000
    0.5000

y =

    2.0000
    2.6900
    3.6181
    4.8663
    6.5451
    8.8032

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