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// example no. 6.21,
// caption: solving the IVP by implicit RK2 method
// u'=f(t,u)
// u'=-2tu^2
//u(0)=1,h=0.2;
t0=0;h=0.2;tn=.4;u0=1;
deff('[z]=f(t,u)','z=-2*t*u^2');
umaxAllowed = 1e+100;
t = [t0:h:tn]; u = zeros(t); n = length(u); u(1) = u0;
for j = 1:n-1
// k1=h*f(t(j)+h/2,u(j)+k1/2);
// conidering the IVP we can infer that the above expression in non linear in k1,
// hence we use newton rapson method to solve for k1;
deff('[w]=F(u2)','w=k1+h*(2*t(j)+h)*(u(j)+k1/2)^2') // u2=u(2)
deff('[x]=Fp(u2)','x=1+h*(2*t(j)+h)*(u(j)+k1/2)')
k1=h*f(t(j),u(j));
newton(k1,F,Fp);
u(j+1) = u(j) +k1
disp(u(j+1))
end;
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