MATLAB: Did I implement the backwards-euler Methode correctly

Question

Hello everyone 🙂

I struggle a little bit with Backwards Euler Method and an excercise our professor gave us:

With h (step size between two adjacent points) = 1/49

Our Task is to plot this. I tried with my Backwards-euler script:

function [t,y] = Euler_backward(t0,T,y0,h,f)
%h = (T-t0)/m ;    %I usually let him calculate h but since it is given I have percentaged(?) it out
t = t0:h:T ;           % my time vector
y = zeros(length(y0), length(t)) ; % To be able to calculate a system, I want it to make a matrix with y0 rows and t columns 
y(:,1) = y0 ;
for i = 2:length(t)-1                                    
ye(:,i) = y(:,i) + h * f(t(i),y(:,i)) ; % forward Euler Method : y(i+1) = y(i) + h*f(t(i),y(i)) to provide y(i+1) for  backwards
y(:,i+1) = y(:,i) + h*f(t(:,i),ye(:,i)) ; % Backwards part with y(i+1) = y(i) + h*f(t(i+1),y(i+1))
end
plot(t,y)
legend('y1','y2')
end

and my separate script (I like to separate script and function):

t0 = 0;
T = 1;
y0 = [1;1];  %Starting conditions
f =@(y,t) [0*y(1)+1*y(2);...    %Made a function from the matrix
    -101*(y1)-100*y(2)];
h = 1/49 ;
Euler_backward(t0,T,y0,h,f)

It then gives me following graph:

and the graph from our professor looks like this:

I didn't graph the solutions but it is pretty clear, that my solution hardly resembles the solution of my professor.

Now I am kinda lost where my mistake is. Did I implement the backwards euler wrong? It would be very kind if someone could help with this 🙂

Have a great day,

Marcus

Answer 1

Hi Marchus,

The backward euler method is implemented correctly, however as f is not a function of t, the plot is constant for increasing t values. Also, f can be updated as shown below.

A = [0 1; -100 -101];
f=@(t,y) A*y;