(*Define variables and functions*)ClearAll[t, x, alpha, s, f]
t = Symbol["t", Positive -> True];
x = Symbol["x", Positive -> True];
alpha = Symbol["alpha", Positive -> True];
s = Symbol["s", Positive -> True];
n = Symbol["n", Integers -> True];
f[n_, x_, t_] := f[n][x, t];

(*Define Elzaki Transform*)
elzakiTransform[func_, var_, transformVar_] := 
 Integrate[
  transformVar*func*Exp[-var/transformVar], {var, 0, Infinity}]

(*Define Inverse Elzaki Transform*)
inverseElzakiTransform[transformFunc_, transformVar_] := 
 InverseLaplaceTransform[transformFunc, transformVar, t]

(*Define the functional correction method*)
correctionFunctional[fPrev_, x_, t_, alpha_] := 
 Module[{delayTerm, rhs, transformedRHS, correctedTransform, 
   correctedFunction},(*Define the delay term*)
  delayTerm = fPrev /. t -> t - 1;

  (*Define the RHS of the equation*)rhs = x^2*t - delayTerm;

  (*Apply Elzaki Transform to RHS*)
  transformedRHS = elzakiTransform[rhs, t, s];

 (*Multiply by s^alpha for fractional derivative correction*)

  correctedTransform = s^alpha*transformedRHS;
  (*Apply inverse Elzaki transform to get back to time domain*)

  correctedFunction = inverseElzakiTransform[correctedTransform, s];
  (*Add the initial condition x^2*)correctedFunction]

(*Example:Initialize f_0(x,t)*)
f0 = x^2;

(*Compute iterations*)
iterations = 3;
fPrevious = f0;

For[i = 1, i <= iterations, i++, 
 fCurrent = correctionFunctional[fPrevious, x, t, alpha];
 Print["Iteration ", i, ":"];
 Print[TraditionalForm[fCurrent]];
 fPrevious = fCurrent;]