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Code help on Euler's method

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Szalka Gergo
Szalka Gergo on 13 Apr 2021 at 18:44
Answered: John D'Errico on 14 Apr 2021 at 17:44
I would like to implement a Matlab code based on Euler's method. This is a project work in the university, and I have a sample solution from my professor to make this project easier. I have succesfully modified this sample solution to fit my task.
Differential equation:
My code:
fv = @(t,y) t * sin(-y) + cos(t)./(t+1);
t0 = 0; tv = 7; y0 = 2; steps = 10;
[tE,yE] = Euler(fv,[t0,tv],y0,steps);
plot(tE,yE,'*;Euler;');
function [t,y] = Euler(fv,tr,y0,steps)
t = linspace(tr(1),tr(2),steps + 1);
y = zeros(size(t));
h = t(2)-t(1);
y(1) = y0;
for i = 1:steps
y(i+1) = y(i) + h * fv(t(i), y(i));
end
end
When I attemp to run my code I always get an error message:
Unrecognized function or variable 'Euler'.
My Matlab skills are horrible, so I can't figure out what is the problem. :(
If anyone provide me an explanation or a proper code, it'll be very helpful for me. Thank you in advance.
  2 Comments
Szalka Gergo
Szalka Gergo on 14 Apr 2021 at 17:12
Thank you for your help. I use MATLAB R2020b (Linux). Euler() is not a separate file, it is located in my main code. I could not solve this issue. Maybe I will try it on another computer.
Meanwhile, I tried to write a new code:
fv = @(t,y) t * sin(-y) + cos(t)./(t+1); % diff. eq.
a = 0; % start point
b = 7; % end point
l = 100; % number of steps
y(1) = 2; % initial condition: y(1) = 2
h = (b-a)/l;
t = a:h:b;
y = zeros(size(t));
% Euler's method
for i = 1:l
y(i+1)= y(i)+ h * fv(y(i),t(i));
end
%ode45 -- it's just a benchmark
[t45,y45] = ode45(fv,[a,b],y(1));
plot(t,y,'b-')
hold on;
plot(t45,y45,'r-')
legend('Euler', 'ode45', 'Location','NorthWest')
hold off;
It's FINALLY working without error message, but I believe that it contain some bugs.
Figure:
It doesn't look too good. The Euler method can't approximate the original DE.

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Accepted Answer

John D'Errico
John D'Errico on 14 Apr 2021 at 17:44
What you need to understand is that Euler is a poor method to use. Easy to write, easy to understand. But is it good? NO. And this is why you should not be writing code to solve numerical methods problems. You want to UNDERSTAND those codes, yes. But then for any real work, you need to use tools like ODE45, ODE15S, etc.
Regardless, look closely at the code you wrote.
fv = @(t,y) t * sin(-y) + cos(t)./(t+1); % diff. eq.
...
y(i+1)= y(i)+ h * fv(y(i),t(i));
Do you see a difference in how you call fv, compared to how you defined the function? You swapped y and t around.

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