ریاضیات

MATLAB code example[edit]

clear; clc; close('all');
y0 = 1;
t0 = 0;
h = 1; % try: h = 0.01
tn = 4; % 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^t):
    tt = (t0:0.001:tn);
    yy = 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\n', 'i', 'yi', 'ti', 'f(yi,ti)');
    fprintf('%10d%+10.2f%+10.2f%+15.2f\n', 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\n', 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 = y;
end

% OUTPUT:
%         i        yi        ti       f(yi,ti)
%         0     +1.00     +0.00          +1.00
%         1     +2.00     +1.00          +2.00
%         2     +4.00     +2.00          +4.00
%         3     +8.00     +3.00          +8.00
%         4    +16.00     +4.00         +16.00
% NOTE: Code also outputs a comparison plot

منبع

https://en.wikipedia.org/wiki/Euler_method