Taylor's Method of Order 2.

The Taylor series method is a numerical method for solving differential equations

%kodi matlabik i serise se teilorit dhe krahasimi i tij me metoden teorike

clc;

clear all;

f = @(t,y) (y-log(t)-2);

fprim=@(t,y) (y-log(t)-2-1/t); % y'=f=y-log(t)-2, f'=y'-1/t+0 =y-log(t)-2-1/t.

a = input('Shkruani fillimin e intervalit, a: ');

b = input('Shkruani fundin e intervalit, b: ');

n = input('Shkruani numrin e nenintervaleve, n: '); %a;h;b

ya = input('Shkruani kushrin fillerstar, ya: ');

h = (b-a)/n;

t=[a zeros(1,n)];

z=[ya zeros(1,n)];

for i = 1:n+1

t(i+1)=t(i)+h;

zprim=f(t(i),z(i))+(h/2)*fprim(t(i),z(i));

z(i+1)=z(i)+h*zprim;

fprintf('%5.4f %11.8f\n', t(i), z(i));

plot(t(i),z(i),'r*-');

grid on;

hold on;

xlabel('t vlera'); ylabel('z vlera');

hold on;

end

syms x y(t);

f= y-log(t)-2;

Dy=diff(y);

ypde=dsolve(Dy== f,'t');

yvde=dsolve(Dy== f,y(a)==ya,'t');

t=linspace(a,b,n);

ysakt=eval(yvde);

plot(t,ysakt,'o-k');