Second Order Runge-Kutta

f = @(t,y) (-12*y);

a = 0;
b = 1;

%z = exp(1);

alpha = 1; %y(0) = 1;

N1 = 20;
N2 = 50;
N3 = 100;

y0 = RK2(f,a,b,alpha,N1).'
y1 = RK2(f,a,b,alpha,N2).';
y2 = RK2(f,a,b,alpha,N3).';

t = 1;

Err_y0 = abs(y0(t)-t)
Err_y1 = abs(y1(t)-t)
Err_y2 = abs(y2(t)-t)

Err = [(Err_y0) (Err_y1) (Err_y2)]

plot(y0,t)
plot(y1,t)
plot(y2,t)
loglog(Err)
hold on

function [y,t] = RK2(f,a,b,alpha,N)

h = (b-a)/N;
t = alpha;
y = alpha;

for i = 1:N
    k1 = h.*f(t,y(i));
    k2 = h.*f(t,y(i)+(k1));

    y(i+1) = y(i)+(k1+k2)/2;
    t = a + i.*h;
end

end