$f≔\left(x\,y\right)\to x\+y^2$

$f:=\left(x\,y\right)\→x\+y^2$

$a≔0$

$a:=0$

${\mathrm{\φ}}_0≔0$

${\mathrm{\φ}}_0:=0$

$N≔3$

$N:=3$

$$\begin{gathered} \mathbf{for} \mathrm{k} \mathbf{from} 0 \mathbf{to} N-1 \mathbf{do} \\ {\mathrm{\φ}}_{k\+1}≔\mathrm{unapply}\left({\mathrm{\φ}}_0\+{\∫}_a^{x}f\left(t\,{\mathrm{\φ}}_k\left(t\right)\right)\ⅆt\,x\right)\; \\ \mathrm{print}\left(\mathrm{nprintf}\left(''Iterate\; number\; \%d:''\,k\+1\right)\right) \\ \mathrm{print}\left({\mathrm{\φ}}_{k\+1}\left(x\right)\right) \\ \mathbf{end} \mathbf{do}\: \end{gathered}$$

$\mathrm{Iterate\; number\; 1:}$

$\frac{1}{2}{x}^2$

$\mathrm{Iterate\; number\; 2:}$

$\frac{1}{2}{x}^2\+\frac{1}{20}{x}^5$

$\mathrm{Iterate\; number\; 3:}$

$\frac{1}{4400}{x}^{11}\+\frac{1}{160}{x}^8\+\frac{1}{20}{x}^5\+\frac{1}{2}{x}^2$