# Mathematica does not show anything after running for higher iterations

Consider a BVP: $$y^{\prime\prime}=\frac{3}{2}y(t)^{2}$$ with the boundary conditions $$y(0)=4$$ and $$y(1)=1$$. The exact solution is $$y(t)=\frac{4}{(1+t)^{2}}$$ and the initial guess is $$y(t)=4-3t$$. Now I want to solve this BVP by an iteration method in mathemetica. The code is given below, which works well for $$n=5$$ but when I put $$n=20$$ it doest show anything after running for some time.

δ = 10^-100;
Clear[x];
x[0] = Function[t, 4 - 3 t];
a[n_] := a[n] = 0.5947894739;
x[n_] := x[n] = Function[t,Evaluate[Chop[Expand[x[n - 1][t]+a[n]*Integrate[s (1 - t) (x[n-1]''[s] - (1.5) x[n - 1][s]^2), {s, 0, t}] +a[n]*Integrate[t (1 - s) (x[n - 1]''[s] - (1.5) x[n - 1][s]^2), {s,t,1}]], δ]]];
NumberForm[a0 = {Table[x[i][0.5], {i, 0, 5}]}]


With an extra Expand before integration, it works:

δ = 10^-100;
a[n_] = 0.5947894739;
Clear[x];
x[0] = Function[t, 4 - 3 t];
x[n_] := x[n] = Function[t, Evaluate[
Chop[Expand[
x[n - 1][t] +
a[n]*Integrate[Expand[s (1 - t) (x[n - 1]''[s] - (1.5) x[n - 1][s]^2)], {s, 0, t}] +
a[n]*Integrate[Expand[t (1 - s) (x[n - 1]''[s] - (1.5) x[n - 1][s]^2)], {s, t, 1}]], δ]]];

Table[x[i][0.5], {i, 0, 20}]
(*    {2.5, 1.76116, 1.77104, 1.77548, 1.77701, 1.77752, 1.77769,
1.77775, 1.77777, 1.77777, 1.77778, 1.77778, 1.77778, 1.77778,
1.77778, 1.77778, 1.77778, 1.77778, 1.77778, 1.77778, 1.77778}    *)

• Dear @Roman, when I calculate absolute error by replacing x[i][0.5] with Abs[x[i][0.5]-1.77778] it shows error 2.22228*10^-6 but in the paper sciencedirect.com/science/article/pii/S0893965918300533 see table2 the error for 20 iteration is 8.075267(-12) for t=0.5 Mar 23 at 15:37
• The correct answer is $16/9$, not $1.77778$. Try x[20][0.5] - 16/9 which gives $-5.40208\times10^{-11}$. Mar 23 at 15:48
• Thanks dear @Roman but this answar is still from the paper I mentioned Mar 23 at 15:54
• Are you sure your a[n] contains all the necessary digits for getting the exact answer from your paper? Mar 23 at 15:58
• Dear @Roman yes I have checked all. The paper is in the link I mentioned above. Mar 23 at 16:03