ImprovedEuler.m

function ImprovedEuler(F)
% Improved Euler Method
% blog.mrfarhad.ir
% author: Farhad Mehryari
a = 0;          
b = 200;

% محدوده مورد نظر

n = 100;            % ـعداد دفعات تکرار حلقه
h = (b - a)/n;      % طول هر مرحله
N(1) = 0.5;         % مقدار اولیه N
t(1) = a;           % مقدار اولیه t

for i=1:n
N(i+1) = N(i) + h*F(N(i),t(i));         
t(i+1) = a + i*h;                       
N(i+1) = N(i) + 0.5*h*(F(N(i+1), t(i+1)) + F(N(i),t(i))) ;    
end

disp(N) % نمایش بردار جواب های هر مرحله
end
% blog.mrfarhad.ir
% author: Farhad Mehryari

usage.m

F = @(N, t) .02*(1 - N/2)*N;
ImprovedEuler(F)
Columns 1 through 8:

   0.50000   0.51515   0.53059   0.54633   0.56235   0.57866   0.59525   0.61210

 Columns 9 through 16:

   0.62922   0.64660   0.66422   0.68208   0.70017   0.71848   0.73700   0.75571

 Columns 17 through 24:

   0.77460   0.79367   0.81289   0.83226   0.85176   0.87137   0.89109   0.91089

 Columns 25 through 32:

   0.93076   0.95069   0.97066   0.99065   1.01065   1.03064   1.05060   1.07053

 Columns 33 through 40:

   1.09039   1.11019   1.12990   1.14951   1.16900   1.18836   1.20757   1.22663

 Columns 41 through 48:

   1.24551   1.26421   1.28271   1.30101   1.31908   1.33693   1.35453   1.37189

 Columns 49 through 56:

   1.38900   1.40583   1.42240   1.43869   1.45470   1.47042   1.48584   1.50097

 Columns 57 through 64:

   1.51580   1.53032   1.54454   1.55846   1.57207   1.58536   1.59836   1.61104

 Columns 65 through 72:

   1.62342   1.63549   1.64726   1.65873   1.66990   1.68078   1.69136   1.70166

 Columns 73 through 80:

   1.71167   1.72139   1.73085   1.74003   1.74894   1.75759   1.76598   1.77412

 Columns 81 through 88:

   1.78201   1.78966   1.79707   1.80424   1.81119   1.81792   1.82443   1.83073

 Columns 89 through 96:

   1.83683   1.84272   1.84842   1.85393   1.85925   1.86439   1.86936   1.87416

 Columns 97 through 101:

   1.87880   1.88327   1.88759   1.89176   1.89578