Solving numerically Pdes and Odes and Iteration processes

Dears, I have the following equations: $$x^{\prime\prime}(t)=\frac{3}{2}x(t)^{2}$$ with $$x(0)=4$$ and $$x(1)=1$$. The exact solution is $$x(t)=\frac{4}{(1+t)^{2}}$$. Now the corresponding operator is $$Px=x+\int_{0}^{t}s(1-t)(x''-\frac{3}{2}x^{2})ds\hspace{1cm} \text{if}\hspace{0.25cm} 0\leq s\leq t$$ and $$Px=x+\int_{t}^{1}t(1-s)(x''-\frac{3}{2}x^{2})ds\hspace{1cm} \text{if}\hspace{0.25cm} t\leq s\leq 1.$$ Now I apply a fixed point scheme on the above operator $$P$$ to find its fixed point: The scheme reads as (where $$x_{0}$$ any initial guess):

$$y_{n}=(1-a_{n})x_{n}+a_{n}Px_{n}, x_{n+1}=Py_{n}.$$

. Now the initial iterate for the above given BVP is $$x_{0}(t)=4-3t$$. For $$t=0.5$$ and $$a_{n}=0.80$$, I write the following code in Mathemtica

Clear[x, P, t, s, a]
x[0] = 4 - 3 t;
P[x_]:=P[x]=Piecewise[{{x+Integrate[s (1 - t) x''[s] - 3/2 (x[s])^2, {s, 0, t}],0<=s<= t},{x+Integrate[(s (1 - t) x''[s] - 3/2 (x[s])^2),{s, t, 1}],t<=s<= 1}}];
a[n_] := a[n] = 0.95;
x[n_] := x[n] = P[(1 - a[n - 1])*x[n - 1] + a[n - 1]*P[x[n - 1]]]
NumberForm[a1 = {Table[x[i] /. t -> (0.5), {i, 0, 20}]}, 7]


The iterations are put 20 as we see above. When I run the code it shows errors. I think it is due to the putting of wrong putting of $$P$$ in the mathemtica. Or there is some additional information's needed to integrate the terms inside the $$P$$. I will be thankful to the kind help please.

• Can you, please, explain how this post is different than this or this both of which were asked by you and have answers?
– kcr
Mar 8 at 19:24
• Dear @Kcr, I want to iterate the operator P directly instead of simplifying. Because this will give me chance to use other iterative methods of the literature directly. Mar 8 at 20:05