I have a numerical differential equation problem called Euler's method. My function is working for a single step size (value h) but I am trying to change the code to allow me to loop over 5 different values h (by changing h from a single value to a list of possible values).

I would like to change h to h=[0.2,0.1,0.05,0.0025,0.0125] and have the values to then create the plot showing the analytical solution and the Euler's method solutions at different values of the step size.

My code that is working for a single value of the step size h is:

So the question is really trying to create Euler's method for different step sizes i.e. "How does one change our function to loop over a list and plot the results with matlab"?

clear; clc; close('all');

y0 = 1;

t0 = 0;

h = 0.2;

tn = 5; % equal to: t0 + h*n, with n the number of steps

[t, y] = Euler(t0, y0, h, tn);

plot(t, y, 'b');

% exact solution y=e^(-1 + e^(-t))):

tt = (t0:0.001:tn);

yy = exp(-1+exp(-tt));

hold('on');

plot(tt, yy, 'r');

hold('off');

legend('Euler', 'Exact');

function [t, y] = Euler(t0, y0, h, tn)

fprintf('%10s%10s%10s%15s ', 'i', 'yi', 'ti', 'f(yi,ti)');

fprintf('%10d%+10.2f%+10.2f%+15.2f ', 0, y0, t0, f(y0, t0));

t = (t0:h:tn)';

y = zeros(size(t));

y(1) = y0;

for i = 1:1:length(t) - 1

y(i + 1) = y(i) + h * f(y(i), t(i));

fprintf('%10d%+10.2f%+10.2f%+15.2f ', i, y(i + 1), t(i + 1), f(y(i + 1), t(i + 1)));

end

end

% in this case, f(y,t) = f(y)

function dydt = f(y, t)

dydt = -exp(-t)*y;

end

Find the true solution of the IVP: y'(t) -e y(t), y(0) = 1. Then write a single MATLAB code in a file named Prob3.m (also upload to Canvas) to preform the following tasks: (a) Using the forward Euler method provided in class, approximate the solution to this problem numerically with stepsizes h = {0.2, 0.1, 0.05, 0.025, 0.0125} for t (0,5). Output the solution and absolute error every 1 unit of time, as well as the overall maximum of these errors for the time interval. (b) Store each of these maximum errors in a vector of the same size as your vector of h values. (c) Using your stored error values, numerically estimate the order of convergence for Euler's method on this problem. (d) Plot numerical solutions vs. the true solution for all h above in the same figure). Find the true solution of the IVP: y'(t) -e y(t), y(0) = 1. Then write a single MATLAB code in a file named Prob3.m (also upload to Canvas) to preform the following tasks: (a) Using the forward Euler method provided in class, approximate the solution to this problem numerically with stepsizes h = {0.2, 0.1, 0.05, 0.025, 0.0125} for t (0,5). Output the solution and absolute error every 1 unit of time, as well as the overall maximum of these errors for the time interval. (b) Store each of these maximum errors in a vector of the same size as your vector of h values. (c) Using your stored error values, numerically estimate the order of convergence for Euler's method on this problem. (d) Plot numerical solutions vs. the true solution for all h above in the same figure)