@Preben Alsholm 

Thank you for quick fix.  :)

@isifesai 

Code for Maple below 2018 version.

Integro-Eq_Ver_3A.mw

@panke

I edited my answer and works as it should.

@Preben Alsholm 

Reading this paper on page 87 says better is Trapezoidal rule.

Download Volterra_integral_equation_ver2.mw

@isifesai 

I speed up computations by numeric integration.Code still requires polishing(Speed up).

Download Integro-Eq_Ver_2A.mw

Works only in Maple 2018 and above.

@mmcdara 

I find a way to solve my problem.

Thank you for your contribution.

Download Volterra_integral_equation.mw

@mmcdara 

Did you read my comments,I need solution with int(numeric) ,not int(symbolic) that I want  CurveFitting for that.

I do not want to wait for computation time an eternity.

Above MMA code give me solution, about a few second.

Thanks anyway.

Volterra2bis_ver2.mw

@sand15 

For func(x,7) it should be :x + x^3/6 + x^5/120 + x^7/5040

for n=7 it should be almost the same as sinh(x).

@sand15 

Thanks you for your efforts,but:

1.At first I need int(numeric),because I have  more complicated example and Int(symbolic) can't solve or is very slow.See

2.it does not matter if I use: Interpolation or CurveFitting.

3.Your answer dosen't work for me.("Problem is this method seems to diverge",Why ? I dosen't know)

Maybe exist a workaround and no need to convert code from MMA to Maple.

MMA solution with more complicated example:

@vv 

Maple's Sum have troubles, at t=1/10, workaround is:

Let's says for: t=3/100.

1/exp(2.)-evalhf(add(eval(2^(-j)*t^(-2*j)*GAMMA(-2*j+1, 1)*exp(-1)/factorial(j), t = 3/100), j = 0 .. 1000));

#-2.296937672*10^237

int(eval(exp(-1-1/v)*(1-exp(v^2/(2*t^2)))/v^2, t = 3/100), v = 0 .. 1, numeric);

#-2.296937672*10^237

,but for  t=2/100 or near t=0  gives: Float(-infinity) :P

Who needs such large numbers?

@weidade37211 

I did some tests and you can see Maple 2017.3 have bad performace and baheves incorrectly. 

@isifesai 

This has nothing to do with the first question.

Please post a new question.

@isifesai 

You can try Mathematica for free online(free plan),only you must sign-in to:https://www.wolframcloud.com

Maplesoft it does not give you that option yet.

@isifesai 

With MMA we can find solution with series and it easy to do: 

n = 5;
yrule = y -> Function[x, Sum[a[j] x^j, {j, 0, n}] + O[x, 0]^(n + 1)];
eq = y'[x] - (Exp[x] + 2/9*Exp[3] - 1/9 + Integrate[s *y[s]^3, {s, 0, 1}]) == 0 /. yrule // Simplify;
le = LogicalExpand[eq];
sol = Solve[le && (y[x] /. yrule /. x -> 0) == 1,Table[a[i], {i, 0, n}]];(*with the initial conditions y[0]\[Equal]1*)
Ysol[x_] = y[x] /. yrule /. First@sol // Normal // N

(* 1. - 5.59113 x + 0.5 x^2 + 0.166667 x^3 + 0.0416667 x^4 + 0.00833333 x^5 *)

Plot[Ysol[x], {x, -1, 1}]

I could translate MMA code to Maple code,but in Maple there is no equivalent to LogicalExpand function ,and this translation(LogicalExpand to Maple) is not my strength.