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How I can obtain $n^{th}$ approximation of the following equation $f(t)=t+\int_0^tds f(s)$ by iteration method?

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  • $\begingroup$ I've rolled back your question to it's original version, as I think your change of dimension is significant enough to warrant a new question. Feel free to open such a new question! Furthermore, no one can really answer this one, as it's closed anyway! (Not that I agree with the closure.) $\endgroup$ Jul 23, 2014 at 14:48
  • $\begingroup$ Please don't roll back my roll back! Honestly, it makes no sense for two reasons. Your original question refers to quite a specific operator so changing the operator wastes the efforts of those who answered the question. Secondly, this question has been closed so no one can answer it now anyway. If you have another question, simply open a new question. $\endgroup$ Jul 23, 2014 at 22:33
  • $\begingroup$ @Javad I encourage you to post a new question as Mark suggested. $\endgroup$
    – Mr.Wizard
    Jul 24, 2014 at 6:37
  • $\begingroup$ I am so sorry @MarkMcClure. I am a beginner in this area. $\endgroup$ Jul 24, 2014 at 11:28
  • $\begingroup$ No biggie - you'll probably be an expert one day! $\endgroup$ Jul 24, 2014 at 11:39

3 Answers 3

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T[f_] := t + Integrate[f /. t -> s, {s, 0, t}];
NestList[T, t, 5]

(* Out: 
  {t, t + t^2/2, t + t^2/2 + t^3/6, 
   t + t^2/2 + t^3/6 + t^4/24, 
   t + t^2/2 + t^3/6 + t^4/24 + t^5/120, 
   t + t^2/2 + t^3/6 + t^4/24 + t^5/120 + t^6/720}
*)

Looks like $f(t)=e^t - 1$ is a fixed point.

T[E^t - 1]
(* Out: -1 + E^t *)
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  • $\begingroup$ what is your opinion about matrix form of the equation? for example $F(t)=A(t)+\int_0^tB(s)F(s)ds$ where F, A and B are matrices. $\endgroup$ Jul 23, 2014 at 0:10
  • $\begingroup$ @JavadKazemi I think the same basic idea should work, though it would be nice to have a specific example in mind. $\endgroup$ Jul 23, 2014 at 1:01
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    $\begingroup$ What is your idea about $A(t)=\bigg{(}\begin{matrix} t & 0 \\ Cos(t) & 1 \end{matrix}\bigg{)}$ and $B(s)=\bigg{(}\begin{matrix} e^s & 0 \\ 0 & e^{-s} \end{matrix}\bigg{)}$? $\endgroup$ Jul 23, 2014 at 14:30
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I assume you wanted the Neumann series method.

ClearAll[f, t, s];
f[0] = t;
f[n_] := f[n] = Integrate[(f[n - 1] /. t -> s) s, {s, 0, t}];
data = Table[f[i], {i, 0, 10}]
Total@data
% /. t -> .5
(*0.543827*)

Mathematica graphics

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  • $\begingroup$ I win - by 10 seconds! :) $\endgroup$ Jul 22, 2014 at 22:57
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Unless you have hidden functions and variables:

n = 4; (*Put whatever term you want here*)
x0 = 0; (*This is the point you want to expand around*)
(*Define g[s] here*)
g[t] = t + Integrate[f[s], {s, 0, t}] ;
Series[g[t], {t, x0, n}];
Normal[%][[n]] (*This is the answer you want*)
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